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TOPOLOGICAL lNVARlANTS OF CONTACT STRUCTURES
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TOPOLOGICAL INVARIANTS OF CONTACT STRUCTURES
AND PLANAR OPEN BOOKS

By

Mehmet Flrat Arlkan

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2008

ABSTRACT

TOPOLOGICAL INVARIANT S OF CONTACT STRUCTURES AND
PLANAR OPEN BOOKS

By

Mehmet Flrat Arlkan

The algorithm given by Akbulut and Ozbagci constructs an explicit open
book decomposition on a contact three-manifold described by a contact
surgery on a link in the three-sphere. In this work, we first improve this
algorithm by using Giroux’s contact cell decomposition process. Our algo-
rithm gives a better upper bound for the recently defined “support genus
invariant” of contact structures. Secondly, we find the complete list of all
contact structures (up to isotopy) on closed three—manifolds which can be
supported by an open book having planar pages with three (but not less)
boundary components. Among these contact structures we also distinguish
tight ones from those which are overtwisted. Finally, we study contact struc-
tures supported by open books whose pages are four—punctured sphere.
Among these contact structures we prove that a certain family is holo—
morphically fillable using lantern relation, and show the overtwistedness of

certain families using the study of right-veering diffeomorphisms.

To my Love and Family

iii

ACKNOWLEDGMENTS

First, I would like to express my sincere gratitude to my supervisor, Selman
Akbulut, for his invaluable guidance, encouragement, and for the motivation
he has given throughout the research. I also would like to thank Onur
Aglrseven, Selahi Durusoy, Cagrl Karakurt, Burak OzbagCI, Arda Bugra
Ozer, and Andras Stipsicz for their suggestions and helpful conversations.
I want to thank my parents for their support.

I am also grateful to my friend Mehmet Oztan for his technical support.
Lastly, but not least, my deepest love and respect to my wife, Setenay
Arlkan, who was the main source of my creativity and supported me all the

way. Without her encouragement, this thesis would not be possible.

iv

TABLE OF CONTENTS

LIST OF TABLES ......................... vii
LIST OF FIGURES ........................ viii
Introduction ............................ 1
1 Preliminaries .......................... 5

1.1 Contact Structures and

open book decompositions ................... 5

1.2 Legendrian knots and contact surgery ............. 9
1.3 Compatibility and stabilization ................ 10
1.4 Symplectic and Stein Fillings ................. 12
1.5 Monodromy and surgery diagrams ............... 13
1.6 Contact cell decompositions .................. 14
1.7 Right-veering diffeomorphisms ................. 15
1.8 Homotopy invariants of contact structures .......... 17
2 An upper bound for the support genus invariant ..... 20
2.1 The algorithm (The proof of Theorem 2.0.5) ......... 26
2.2 Examples ............................ 43

3 Planar contact structures with binding number three . . 51

3.1 The proof of Theorem 3.0.2 .................. 58
3.2 The proof of Theorem 3.0.4 .................. 62
3.3 Remarks on the remaining cases ................ 86

4 Contact structures associated to

four-punctured sphere ..................... 90
4.1 Four—punctured sphere .................... 92
4.2 Holomorphically fillable contact structures .......... 95

4.3 Overtwisted contact structures ................. 96
4.4 The proof of Theorem 4.0.2 .................. 101

BIBLIOGRAPHY ......................... 106

vi

3.1
3.2
3.3

3.4

3.5
3.6
3.7
3.8

3.9

LIST OF TABLES

All planar contact structures on S3 with binding number 3 3. . 61
Thecaser=0(|p|<2or|q|<2). ............... 67
The case 1” =1, Ipl < 2 or |q| < 2 (and the case (p,q,r) =

(—2,q,1)) ............................. 69

The case r = —1,|p|< 2 or |q| < 2 (and the case (p,q,r) =

(2,q,—1)) ............................. 71
Thecaser=2(|p|<2or|q|<2). ............... 73
Thecaser=—2(|p|<2or|q|<2). .............. 76
Thecaser>2(|p|<2or|q|<2). ............... 82
Thecaser<—2(|p|<2or|q|<2). .............. 83

The case r=0,:l:1, |p| 22, |q| 22 (bn(§) =3 in each row). . 87

vii

LIST OF FIGURES

1.1 Mapping torus S(0ha_1), before and after the cyclic permutation. 8

1.2 Lifts of a and ,3 in the universal cover S. ............ 16

2.1 The square bridge diagram D for the Legendrian right trefoil. . 21

2.2 Constructing a new contact cell decomposition. ......... 25
2.3 The region P for the right trefoil knot and its division ...... 27
2.4 An arbitrary element R2 in ‘13. .................. 28
2.5 The region P for the right trefoil knot ............... 30

2.6 (a) The Legendrian unknot ’71:, (b) the ribbon Fk, (c) the disk

Dk (shaded bands in (b) are the bands ch’s). ........ 32
2.7 Replacing R, R’ with their union R” . .............. 40
2.8 Modifying the algorithm for the case when L is split. ...... 42
2.9 Legendrian right trefoil knot sitting on F55. ........... 45

2.10 (a) The page F for the right trefoil knot, (b) construction of A
and saddle disks. ........................ 46
2.11 (a),(b) Legendrian figure-eight knot, (c) the region P. ..... 47

2.12 Modifying the region P. ...................... 48

viii

2.13 The figure-eight knot on F55. ...................
2.14 (a) The page F for the figure-eight knot, (b) construction of A
and saddle disks. ........................

3.1 The surface 2 and the curves giving the generators of Aut(2, 82).

3.2 Seifert fibered manifold Y(p, q, r) ..................
3.3 Contact manifold (Y(p,q,r),€p,q,,.). ................
3.4 The are a and its image 45(0) = Daprq(a) ............
3.5 Contact surgery diagram corresponding to (Z, abpw). . . . . . .
3.6 Smooth surgery diagram corresponding to Figure 3.3 .......
3.7 Sequence of blow-ups and blow—downs. ..............
3.8 Contact surgery diagrams for (Y, 77). ...............
a E_1,q,0 on S3#L(IqI, 1) z L(IqI,1), (b) the link 14-1,“).

3.10 a 6-2,,“ on L Iq+2I,1), (b) the link 111-24,] ...........

3.9

(
a £03,171 on L(|q I, 1,) (b )the l1nk lLOq 1. ............
(

3.11

3.12 a g-1.q,2 on L Iq — 2|,1) for q S —1, (b) the link lL_1,q,2.

3.13 a £1,q,_2 on L(Iq + 2|, 1) for q < —3, (b) the link L1,q,_2.

3.14

3.15 a fo,_1,_2 on S3#L(2, 1) z L(2,1), (b) the link 1407172.

3.16 a €4,272 on L(4,1), (b) the link 114-1372. ...........

( )
( )
( )
( )
( )
(a) €0.1.—2 on S3#L(2, 1) z L(2,1), (b) the link L0,1,_2 ......
( )
( )
3.17 (a) gm. on L(IrI, 1) for r < —2, (b) the link 1.1m ........
( )

3.18 a an-” on L(IrI, 1) for r < —2, (b) the link 1140:”. .....

4.1 Four—punctured sphere Z, and the simple closed curves. . . . .

4.2 12(7) is to the left of 7 (Left and right sides are identified). . .

53
54
54
57
59
59
60
63
68
70
72
75
77
78
80
81
84

85

91

4.3

4.4

4.6

4.7

D5(h('y)) is to the left of 7-" (Left and right sides are identified). 97

The point 61(1) 6 LC,O :2: IR, and how it is mapped to the right of

itself ................................ 101
The curves 0;, and their images under d)’ in Z. ......... 102
The curves 6;, and their images under qb’ in 2. ......... 103
The curve 7 and its images under two possible gb’ in E. . . . . 104

Introduction

In the past decade, the study of open book decompositions (open books) in
contact geometry and related fields became more and more important in or-
der to understand the connection between geometry and topology. In 1975,
Thurston and Winkelnkemper [TWI constructed a special contact structure
on a given closed 3-manifold equipped with an open book decomposition
in the sense that certain compatibility conditions are satisfied. In 2002,
Giroux [Ci] proved the existence of a 1 — 1 correspondence between open
books and compatible contact structures on a fixed closed 3-manifold mod-
ulo certain equivalence relations. Based on this correspondence, in 2002,
Ozsvath and Szabo [OSz] defined Heegaard Floer contact invariant (also
called OS—contact invariant) using open books. OS —invariant has allowed
for a much better understanding of tight but not fillable contact structures.
It is nonzero for a Stein fillable structure and it vanishes for an overtwisted
one. In particular, if the invariant is nonzero, then the contact structure is
tight. Later (in 2005) Honda, Kazez and Matic [HKMlI initiated the study
of right-veering diffeomorphisms of surfaces with boundary. They proved
that a contact structure on a closed 3-manifold is tight if and only if the

monodromy map of any compatible open book is right—veering. Recently

(in 2006), Etnyre and Ozbagci [E0] introduced two purely topological in-
variants of a given contact structure based on the above correspondence.

Let (M, 5) be a closed oriented 3-manifold with the contact structure 5, and
let (S, h) be an open book (decomposition) of M which is compatible with
5. In this case, we also say that (S, h) supports 5 (for the definitions see
the next chapter). Based on Giroux’s correspondence theorem (Theorem

1.3.2), we can ask two natural questions:
(1) What is the possible minimal page genus 9(S) =genus(S)?

(2) What is the possible minimal number of boundary components of a page

S with 9(S) minimal?

In IE0], two topological invariants 39(5) and bn(5) were defined to be the

answers. More precisely, we have:

89(5) 2 min{ 9(S) I (S, h) an open book decomposition supporting 5},

called the support genus of 5, and

lm(5) = min{ IBSI I (S, h) an open book supporting 5 and 9(S) = 39(5)},

called the binding number of 5. There are some partial results about these

invariants. For instance,

Theorem 0.0.1 ([EtII) If (M, 5) is overtwisted, then 39(5) = 0. Cl

Also the contact structures with support genus zero and the binding number

2

less than or equal to two are classified in [EC] (see Theorem 3.0.1). Unlike
the overtwisted case, there is not much known yet for 39(5) when 5 is
tight. On the other hand, if we, furthermore, require that 5 is Stein fillable,
then an algorithm to find an open book supporting 5 was given in [A0].
Although their construction is explicit, the pages of the resulting open books
arise as Seifert surfaces of torus knots or links, and so this algorithm is far
from even approximating the number 39(5). In [St], the same algorithm
was generalized to the case Where 5 need not to be Stein fillable (or even
tight), but the pages are still of large genera.

Based on the result of Ding and Geiges [DC] (2004), and by the result of
Gompf [Cm] (1998) on Stein surfaces, we can study the contact structures
and their possible fillings by using Legendrian framed link diagrams de-
scribing them. It is not clear, in general, yet that what property of such
a. diagram determines the tightness or the type of the filling. However,
once we obtain a compatible open book, we can encode the surgery data
by means of its monodromy, and then the monodromy can determine the
tightness and the type of the filling for some (and maybe for all) such di-
agrams. To study overtwistedness, right-veering diffeomorphisms are very
handy. On the other hand, OS—contact invariant and its recent interpre-
tation, EH —-invariant (defined in [HKM2]), can be used to determine the
tightness for some cases.

As an another approach, we can fix an abstract surface and study all possible
monodromies to understand which ones give tight structures and which ones
not. Planar surfaces (the surfaces of genus zero) are the best choices to start

3

with for such an approach. By Theorem 0.0.1, if 39(5) 7é 0, then 5 is tight.
Therefore, understanding planar open books almost solves the tightness
classification of contact structures. We remark that a monodromy which is
not a right—veering can not support a tight structure. In fact, the following

conjecture is the main motivation of the author throughout this work.

Conjecture 0.0.2 If the monodromy of an open book with minimal page
genus and binding number is right-veering, then the compatible contact

structure is tight.

This dissertation thesis is organized as follows: In the preliminaries chapter
(Chapter 1), we will give the basic definitions and facts which will be used
in the rest of the work. In Chapter 2, we will present an algorithm which
finds a reasonable upper bound for 39(5) using the given contact surgery
diagram of 5. We will also give examples to understand how efficient the
new algorithm is. In Chapter 3, the complete list of contact structures with
39(5) 2 0 and bn(5) = 3 will be given by analyzing the mapping class group
of the surface so called “pair of pants”. In Chapter 4, we will explore the
planar contact structures with bn(5) g 4: After studying the mapping class
group of “four—punctured sphere”, we will prove that a certain family is
holomorphically fillable, and also show the overtwistedness for some other

family.

Chapter 1

Preliminaries

This chapter provides the definitions, notation and facts that will be used in
the next chapters. Some well—known material is included in order to make

the presentation self-contained.

1.1 Contact Structures and

open book decompositions

A 1-form 0: E 91(M) on a 3-dimensional oriented manifold M is called a
contact form if it satisfies a /\ da 7Q 0. An oriented contact structure on
M is then a hyperplane field 5 which can be globally written as the kernel
of a contact 1-form a. We will always assume that 5 is a positive contact
structure, that is, a A do > 0. Note that this is equivalent to asking that
do be positive definite on the plane field 5, ie., dalg > 0. Two contact
structures 50, 51 on a 3-manifold are said to be isotopic if there exists a 1-
parameter family 51 (0 S t S 1) of contact structures joining them. We say
that two contact 3-manifolds (ll/11,51) and (Mg,52) are contactomorphic if
there exists a diffeomorphism f : MI ——> Mg such that f..(51) = 52. Note

5

that isotopic contact structures give contactomorphic contact manifolds by
Gray’s Theorem, and any contact 3—manifold is locally contactomorphic
to (R3,50) where standard contact structure 50 on R3 with coordinates
(33,9,z) is given as the kernel of a0 = dz + xdy. The standard contact
structure £3: on the 3-sphere S3 = {(r1,r2,61,62) : r? + r3 = 1} C C2 is
given as the kernel of ast = riddl + rgddg. One basic fact is that (R3, 50) is
contactomorphic to (S3 \ {pt}, 5st). For more details on contact geometry,
we refer the reader to [Ge], [Et3I, and [MS].

An open book decomposition of a closed 3-manifold .M is a pair (L, f)
where L is an oriented link in M, called the binding, and f : M \ L —-+ S1
is a fibration such that f ‘1(t) is the interior of a compact oriented surface
St C M and 6S, 2 L for all t 6 S1. In such a case, L is also called a
fibered link in M. The surface S = St, for any t, is called the page of
the open book. The monodromy of an open book (L, f) is given by the
return map of a flow transverse to the pages (all diffeomorphic to S) and
meridional near the binding, which is an element h E Aut(S, BS) , the group
of (isotopy classes of) diffeomorphisms of S Which restrict to the identity
on 8S . The group Aut(S,BS) is also said to be the mapping class group
of S, and denoted by F(S).

An open book can also be described as follows. First consider the mapping

torus

5(h) = [0,1] x S/(1,a:) ~ (0,h($))

where S is a compact oriented surface with n = IBS I boundary components

and h is an element of Aut(S,8S) as above. Since h is the identity map
on OS , the boundary 6S (h) of the mapping torus S (it) can be canonically
identified with n copies of T2 = S1 x S1, where the first S1 factor is
identified with [0, 1] / (0 ~ 1) and the second one comes from a component
of BS. Now we glue in n copies of D2 x S 1 to cap off S (h) so that 6D2 is
identified with S1 = [0, 1] / (0 ~ 1) and the S1 factor in 02 x S 1 is identified

with a boundary component of 3S . Thus we get a closed 3-manifold
M = MW, 2: S(h) on D2 x 31

equipped with an open book decomposition (S, h) whose binding is the union
of the core circles in the D2 x S1 ’s that we glue to S (h) to obtain M. To
summarize, an element h E Aut(S, 88) determines a 3—manifold M =
M(S.h) together with an “abstract” open book decomposition (S, h) on it.
For further details on these subjects, see [Cd], and [Et2].

Throughout the work, D7 will denote the right Dehn twist along the sim-
ple closed curve 7, and most of the time we will write 7 instead D, for
simplicity. We want to state the following classical fact which will be used
in Section 4.1. We also give a proof since the author couldn’t find the given

version of the theorem in the literature.

Theorem 1.1.1 Let S be any surface with nonempty boundary, and let

0,h E Aut(S, 8S). Then there exists a contactomorphism

(M(s.h>v€<s.h>l '5 (M(S,aho—1)?€(S,aho'1))'

Proof: The proof based on the idea of breaking up the monodromy
(rho‘1 into pieces as depicted in Figure 1.1. First take each glued solid
torus (around each binding component) out from both (M(3,h),5(5,h)) and
(MIS,0M_1),5IS,UM_1)) to get the mapping tori (S, h) and S(oho‘l). By
breaking the monodromy (rho—1, the mapping torus S (oho’l) = [0,1] x
S/ (1, :13) ~ (0,0ho"1(a:)) can be constructed also as follows: We write

4

swim-1) = (1153/ ~.

i=1

IN.

where S,- = S x [Lg—1, 4I and ~ is the equivalence relation that glues S x {i}
in S1 to S x {i} in S2 by a, glues S x g} in S2 to S x G} in S3 by
h, glues S x {311} in S3 to S x {g} in S4 by 0'1, glues S x {1} in S4 to
S x {0} in S1 by the identity map id. (See the picture on the left in Figure
1.1.)

ids—s l] a U E

Figure 1.1. Mapping torus S (aha—1), before and after the cyclic permutation.

 

 

Since S (oho‘l) is a fiber bundle over the circle S1, we are free to change
its monodromy by any cyclic permutation. Therefore, the monodromy

element 0—1 - id - oh = h also gives the same fiber bundle S (aha—1) (the

8

picture on the right in Figure 1.1 shows the new configuration of S (aha-l)
after the cyclic permutation). Therefore, S (h) = S (oho’l). By gluing
all solid tori back, we conclude that (M(3,h),5(3,h)) is contactomorphic to

(M(S,aho‘1)’€(S,aha‘1)) ° D

1.2 Legendrian knots and contact surgery

A Legendrian knot K in a contact 3-manifold (M,5) is a knot that is
everywhere tangent to 5. Any Legendrian knot comes with a canonical
contact framing (or Thurston-Bennequin framing), which is defined by a
vector field along K that is transverse to 5. We call (M ,5) (or just 5)
overtwisted if it contains an embedded disc D m D2 C M with boundary
8D z S1 a Legendrian knot whose contact framing equals the framing it
receives from the disc D. If no such disc exists, the contact structure 5 is
called tight.

For any p,q E Z, a contact (r)-surgery (r = p/q) along a Legendrian
knot K in a contact manifold (M ,5) was first described in [DC]. It was
proved in [Ho] that if r = 1/k with k E Z, then the resulting contact
structure is unique up to isotopy. In particular, a contact (in-surgery
along a Legendrian knot K on a contact manifold (M,5) determines a
unique surgered contact manifold which will be denoted by (M, 5 )( K.i1)-

The most general result along these lines is:

Theorem 1.2.1 ([DG]) Every closed contact 3-manifold (M,5) can be

obtained via contact (:tl)-surgery on a Legendrian link in (S3,53t). [3

Any closed contact 3-manifold (M, 5) can be described by a contact surgery
diagram drawn in (R3, 50) C (S3, 53¢). By Theorem 1.2.1, there is a contact
surgery diagram for (M, 5) such that the contact surgery coefficient of any
Legendrian knot in the diagram is i1. For any oriented Legendrian knot
K in (R3, 50), we compute the Thurston-Bennequin number tb(K), and the

rotation number rot(K) as

tb(K) = bb(K) — (# of left cusps of K),

1
rot(K) = §[(# of downward cusps) — (# of upward cusps)]

where bb(K) is the blackboard framing of K.
If a contact surgery diagram for (M, 5) is given, we can also get the smooth
surgery diagram for the underlying 3-manifold M . Indeed, for a Legendrian

knot K in a contact surgery diagram, we have:
Smooth surgery coeff. of K 2 Contact surgery coeff. of K + tb(K)

For more details see [OSt] and [Gm].

1.3 Compatibility and stabilization

A contact structure 5 on a 3-manifold M is said to be supported by an open
book (L, f) if 5 is isotopic to a contact structure given by a 1-form a such

that

10

1. do is a positive area form on each page S m f ‘1(pt) of the open book

and
2. a > 0 on L (Recall that L and the pages are oriented.)

When this holds, we also say that the open book (L, f) is compatible with

the contact structure 5 on M.

Definition 1.3.1 A positive (resp., negative) stabilization S};(S, h) (resp,
SI}(S, h) ) of an abstract open book (S, h) is the open book

1. with page S’ = S U I-handle and

2. monodromy h’ = h 0 DK (resp., h’ = h o D]? ) where DK is a right-
handed Dehn twist along a curve K in S’ that intersects the co-core

of the I-handle exactly once.

Based on the result of Thurston and Winkelnkemper [TW] which introduced
open books into the contact geometry, Giroux proved the following theorem

strengthening the link between open books and contact structures.

Theorem 1.3.2 ([GiI) Let M be a closed oriented 3-manifold. Then
there is a one-to—one correspondence between oriented contact structures
on M up to isotopy and open book decompositions of M up to positive
stabilizations: Two contact structures supported by the some open book are
isotopic, and two open books supporting the same contact structure have a

common positive stabilization. El

11

For a given fixed open book (S, h) of a 3-manifold Il/I , there exists a unique
compatible contact structure up to isotopy on M = M(S,h) by Theorem
1.3.2. We will denote this contact structure by 5(3),). Therefore, an open
book (S, h) determines a unique contact manifold (M(S.h): 5(5,h)) up to con-
tactomorphism. We will shorten the notation as (Mh, 5h) if the surface S
is clear from the content.

Taking a positive stabilization of (S, h) is actually taking a special Murasugi
sum of (S, h) with the positive Hopf band (H +, D,) where ”y C H + is the
core circle. Taking a Murasugi sum of two open books corresponds to taking
the connect sum of 3-manifolds associated to the open books. The proofs

of the following facts can be found in [Gd], [Et2].
Theorem 1.3.3 We have

(Ms;(3,h)a€3;;(s,h)) '5 (M(3,h).€(s,h))#(53,€st) ’5 (M(3.h),€(3,h)l- '3

1.4 Symplectic and Stein Fillings

A contact manifold (M , 5 ) is called symplectically fillable if there is a com-
pact symplectic 4—manifold (X, to) (that is, w is a non—degenerate and closed
2—form on X) such that 8X = M and wlg 7E 0. A Stein manifold is a
triple (X, J, to) where J is a complex structure en X, w : X —> IR, and the
2—form w¢ = —d(dw o J) is non-degenerate. We say (M, 5) is called Stein
(holomorphically) fillable if there is a Stein manifold (X, J, 19) such that w
is bounded from below, M is a non-critical level of w and —(di,b o J) is a
contact form for 5. We note that any Stein filling (X, J, 11)) of (M, 5) can

12

be also considered as the symplectic filling (X ,wy) See [Et2] or [OSt] for

the related facts. We will use the following theorem later.

Theorem 1.4.1 ([EGI) Any symplectically fillable contact structure is

tight. ( => Any holomorphically fillable contact structure is tight. ) Cl

Following fact was first implied in [LP], and then in [A0]. The given version

below is due to Giroux and Matveyev. For a proof, see [OSt].

Theorem 1.4.2 A contact structure 5 on .M is holomorphically fillable if
and only if 5 is supported by some open book whose monodromy admits a

factorization into positive Dehn twists only. El

1.5 Monodromy and surgery diagrams

Given a contact surgery diagram for a closed contact 3-manifold (M, 5),
we want to construct an open book compatible with 5. One implication
of Theorem 1.2.1 is that one can obtain such a compatible open book by
starting with a compatible open book of (S3,53t), and then interpreting
the effects of surgeries (yielding (M, 5) ) in terms of open books. However,
we first have to realize each surgery curve (in the given surgery diagram
of (M ,5)) as a Legendrian curve sitting on a page of some open book
supporting (S3, 5st). We refer the reader to Section 5 in [Et2] for a proof of

the following theorem.

13

Theorem 1.5.1 Let (S, h) be an open book supporting the contact manifold

(M ,5). If K is a Legendrian knot on the page S of the open book, then

(Ma€)(K,il) = (M(s,hoo,f,),€(s,hoo;))- D

1.6 Contact cell decompositions

The exploration of contact cell decompositions in the study of open books
was originally initiated by Gabai [Ga], and then developed by Giroux [Gi].
We want to give several definitions and facts carefully.

Let (M, 5) be any contact 3—manifold, and K C M be a Legendrian knot.
The twisting number tw(K, Fr) of K with respect to a given framing Fr
is defined to be the number of counterclockwise 27r twists of 5 along K,
relative to Fr. In particular, if K sits on a surface S C M, and F r5 is the
surface framing of K given by S, then we write tw(K, S) for tw(K , F r3).
If K = 83, then we have tw(K, S) = tb(K) (by the definition of tb).

Definition 1.6.1 A contact cell decomposition of a contact 3- manifold

(M, 5) is a finite C W—decomposition of M such that
(I) the I-skeleton is a Legendrian graph,

(2) each 2-cell D satisfies tw(BD, D) = —1, and
(3) 5 is tight when restricted to each 3—cell.

Definition 1.6.2 Given any Legendrian graph G in (M ,5), the ribbon of

G is a compact surface R = R0 satisfying

14

1. R retracts onto G,
2. TpR 2 5p for all p E G, and

3. TpR7E5p for all p6 R\G.
For a proof of the following theorem we refer the reader to [Cd] and [Et2].

Theorem 1.6.3 Given a closed contact 3— manifold (M ,5), the ribbon of
the I-skeleton of any contact cell decomposition is a page of an open book

supporting 5 . E]

1.7 Right-veering diffeomorphisms

We recall the right—veering diffeomorphisms originally introduced in
[HKM1I. If S is a compact oriented surface with 8S 79 0, the sub-
monoid Veer(S,8S) of right-veering elements in Aut(S,BS) is defined
as follows: Let or and 6 be isotopy classes (relative to the endpoints) of
properly embedded oriented arcs [0,1] —-> S with a common initial point
(1(0) 2 6(0) = :1: 6 (9S. Let 7r : S ——> S be the universal cover of S (the
interior of S will always be R2 since S has at least one boundary compo-
nent), and let :7: 6 8S be a lift of a: 6 8S. Take lifts a and S of a and
B with 61(0) 2 8(0) 2 it. 62 divides S into two regions — the region “to
the left” and the region “to the right”. We say that 5 is to the right of a,
denoted a 2 [3, if either a = 6 (and hence 5(1) 2: 3(1)), or 3(1) is in the
region to the right (Figure 1.2).

15

”the region the region
_ to the left to the right

'\ /

are) = = 8(0)

HM

Figure 1.2. Lifts of a and 6 in the universal cover S.

As an alternative to passing to the universal cover, we first isotop a and fl ,
while fixing their endpoints, so that they intersect transversely (this include
the endpoints) and with the fewest possible number of intersections. Then 6

is to the right of a if the tangent vectors (6(0), 02(0)) define the orientation

onSat :13.

Definition 1.7.1 Let h : S —> S be a difieomorphism that restricts to the
identity map on 83 . Let a be a properly embedded oriented are starting at
a basepoint :1: 6 8S. Then h is right-veering ( that is, h E Veer(S,8S))
if for every choice of basepoint a: 6 (9S and every choice of 0 based at as,
h(a) is to the right of Or (at 2:). If C is a boundary component of S, we
say is h is right-veering with respect to C if h(a) is to the right of a for

all a starting at a point on C.

It turns out that Veer(S, OS) is a submonoid and we have the inclusions:
Dehn+(S,8S) C Veer(S,8S) C Aut(S,8S).

We will use the following two results of [HKMl].

16

Theorem 1.7.2 ([HKM1]) A contact structure I M ,5) is tight if and only
if all of its compatible open book decompositions (S, h) have right-veering

h E Veer(S,8S) C Aut(S,8S). E]

1.8 Homotopy invariants of contact structures

The set of oriented 2—plane fields on a given 3-manifold III is identified
with the space Vect(M) of nonzero vector fields on M. v1,v2 E Vect(M)
are called homologous (denoted by v1 ~ v2) if v1 is homotopic to v2 in M \B
for some 3—ball B in M. The space Spinc(M) of all spinc structures on
M is the defined to be the quotient space Vect(M)/ ~. Therefore, any
contact structure 5 on 1W defines a spine structure tg E Spinc(M) which
depends only on the homotopy class of 5. As the first invariant of 5,
we will use the first Chern class c1(5) E H 2(M ;Z) (considering 5 as a
complex line bundle on M). For a spine structure t5, whose first Chern
class c1(t§)(:= c1(5)) is torsion, the obstruction to homotopy of two 2-
plane fields (contact structures) both inducing t5 can be captured by a
single number. This obstruction is the 3-dimensional invariant d3(5) of
5). To compute d3(5), first recall an almost complex manifold is a pair
(X, J) where J : TX —> TX is such that J2 = —idTX. Now suppose that a
compact almost complex 4-manifold (X, J) is given such that 6X 2 M, and
5 is the complex tangencies in TM, ie., 5 = TMflJ(TM). Let o(X), x(X)

denote the signature and Euler characteristic of X, respectively. Then we

17

have

Theorem 1.8.1 ([GmI) If c1 (5) is a torsion class, then the rational num-

ber

age) = ism. J) — 3a<X> — 2x00)

is an invariant of the homotopy type of the 2-plane field 5. Moreover, two
2-plane fields 5 and 77 with t5 = t,7 and c1 (5) = c1(n) a torsion class are

homotopic if and only if d3(5) = d3(77). D

As a result of this fact, if (M ,5) is given by a contact (in-surgery on a

link sitting in (S3,53t), then we can compute d3(5) using

Corollary 1.8.2 ([DGSI) Suppose that (M,5), with c1(5) torsion, is
given by a contact (:l:1)-surgery on a Legendrian link IL C (53,535) with
tb(K) 75 0 for each K C IL on which we perform contact (+1)-surgery. Let
X be a 4-manifold such that 6X : A/I . Then

are = fie? — 30m — 2x00) + s,

where 3 denotes the number of components in IL on which we perform
(+1)-3urgery, and c E H2(X;Z) is the cohomology class determined by
C(ZK) = rot(K) for each K C IL, and [Ex] is the homology class in
H2(X) obtained by gluing the Seifert surface of K with the core disc of the

2-handle corresponding K. E]

18

We use the above formula as follows: Suppose IL has k components. Write
IL = UikKi. By converting all contact surgery coefficients to the topological
ones, and smoothing each cusp in the diagram, we get a framed link (call it
IL again) describing a simply connected 4-manifold X such that 6X = M.

Using this description, we compute
x(X) 2 1+ k, and o(X) = 0(AL)

where AL is the linking matrix of IL. Using the duality, the number 02 is

computed as
c2 = (PD(C))2 = lbl b2 ° “kaAIlel 1’2"")le

where PD(c) E H2(X,0X;Z) is the Poincaré dual of c, the row matrix

[b1 b2 - ' - by] is the unique solution to the linear system
ALIbl b2 - - - kaT = [rot(Kl) TOL(K2) - - ~rot(Kk)]T.

Here the superscript “T ” denotes the transpose operation in the space of

matrices. See [DGS], [Gm] for more details.

19

Chapter 2

An upper bound for the support

genus invariant

In this chapter, we will present an explicit construction of a supporting open
book (with considerably less genus) for a given contact surgery diagram of
any contact structure 5. Of course, because of Theorem 0.0.1, our algo-
rithm makes more sense for the tight structures than the overtwisted ones.
Moreover, it depends on a choice of the contact surgery diagram describing
5. Nevertheless, it gives better and more reasonable upper bound for 39(5)
(when 5 is tight) as we will see from our examples in Section 2.2.

Let L be any Legendrian link given in (R35) 2 ker(ao 2 dz + xdy)) C
(S3,5,t). L can be represented by a special diagram D called a square
bridge diagram of L (see [Ly]). We will consider D as an abstract diagram

such that

1. D consists of horizontal line segments h1,...,hp, and vertical line seg-

ments v1, ...,vq for some integers p _>_ 2, q 2 2,
2. there is no collinearity in {h1,. . . , hp}, and in {v1, . . . ,vq}.

20

3. each h,- (resp., each vj) intersects two vertical (resp., horizontal) line

segments of D at its two endpoints (called corners of D), and

4. any interior intersection (called junction of D) is understood to be a

virtual crossing of D where the horizontal line segment is passing over

the vertical one.

We depict Legendrian right trefoil and the corresponding D in Figure 2.1.

 

D

’05

/T\\¥/

’U4

 

 

 

 

 

 

Legendrian right trefoil

 

 

 

a
m
”U3 ’U2
h3
M
m
p=q=5

v1

Figure 2.1. The square bridge diagram D for the Legendrian right trefoil.

Clearly, for any front projection of a Legendrian link, we can associate a

square bridge diagram D. Using such a diagram D, the following two

facts were first proved in [A0], and later made more explicit in [P1]. Below

versions are from the latter:

Lemma 2.0.3 Given a Legendrian link L in (R3,50), there exists a torus

link Tm (with p and q as above) transverse to 50 such that its Seifert

21

surface FM contains L, dog is an area form on FM, and L does not

separate FM . El

Proposition 2.0.4 Given L and Fm as above, there exist an open book

decomposition of S3 with page Fm such that:
1. the induced contact structure 5 is isotopic to 50;

2. the link L is contained in one of the page FM, and does not separate

it;
3. L is Legendrian with respect to 5;

4. there exist an isotopy which fixes L and takes 5 to 50, so the Legendrian

type of the link is the same with respect to 5 and 50;

5. the framing of L given by the page FM of the open book is the same

as the contact framing. [:1

Being a Seifert surface of a torus link, Fm is of large genera. In Section
2.1, we will construct another open book (98 supporting (S3, 53:) such that
its page F arises as a subsurface of Fm (with considerably less genera),
and given Legendrian link L sits on F as how it sits on the page Firm of
the construction used in [AC] and [PI]. The page F of the open book
(93 will arise as the ribbon of the 1-skeleton of an appropriate contact cell

decomposition for (S3, 58;). As in [Pl], our construction will keep the given

22

link L Legendrian with respect to the standard contact structure 53,. The

following theorem summarize our algorithm:

Theorem 2.0.5 Given L and Fm as above, there exists a contact cell

decomposition A of (S3,53t) such that

1. L is contained in the Legendrian I—skeleton G of A.

2. The ribbon F of the 1-skeleton G is a subsurface of FM

{p and q as above).
3. The framing of L coming from F is equal to its contact framing tb(L).

4. pr > 3 and q > 3, then the genus 9(F) of F is strictly less than the

genus 9(FM) of Fm-

As an immediate consequence (see Corollary 2.1.1), we get an explicit de-
scription of an open book supporting (S3,5) whose page F contains L
with the correct framing. Therefore, if (M i, 5*) is given by contact (:l:1)-
surgery on L (such a surgery diagram exists for any closed contact 3-
manifold by Theorem 1.2.1), we get an open book supporting 5i with page
F by Theorem 1.5.1. Hence, 9(F) improves the upper bound for 39(5) as
9(F) < 9(Fp,q) (for p > 3, q > 3). It will be clear from our examples in
Section 2.2 that this is indeed a good improvement.

The following lemma will be used in the next section.

Lemma 2.0.6 Let A be a contact cell decomposition of a closed contact 3-
manifold (M ,5) with the 1—3keleton G. Let U be a 3-cell in A. Consider
two Legendrian arcs I C 8U and J C U such that

23

1. I c G,
2. J08U=6J=8L
3. G = I U3 J is a Legendrian unknot with tb(G) = —1.

Set G’ = G U J. Then there exists another contact cell decomposition A’

of (M, 5) such that G’ is the I-skeleton of A’

Proof: The interior of the 3—cell U is contactomorphic to (R3,50).
Therefore, there exists an embedded disk D in U such that 8D = C’ and
int(D) C int(U) as depicted in Figure 2.2(a). We have tw(le, D) = —1
since tb(G) = —1. As we are working in (R3, 50), there exist two C°°-small
perturbations of D fixing SD = C such that perturbed disks intersect each
other only along their common boundary C. In other words, we can find

two isotopies H1, H2 : [0,1] x D ——) U such that for each i = 1,2 we have
1. H,(t, .) fixes (9D = G pointwise for all t E [0,1],
2. Hi(0, D) 2 [dB where [(11) is the identity map on D,

3. H,(1, D) = D,- where each D,- is an embedded disk in U with int(Di) C
int(U),

4. D H D1 0 D2 = C’ (see Figure 2.2(b)).

Note that tw(BD,,D,-) = tw(G, Di) 2 —1 for i = 1,2. This holds because
each D,- is a small perturbation of D, so the number of counterclockwise

twists of 5 (along K) relative to F r1), is equal to the one relative to F rD.

24

 

 

 

 

 

.....................

 

............................................

Figure 2.2. Constructing a new contact cell decomposition.

Next, we introduce G’ = G U J as the l-skeleton of the new contact cell
decomposition A’. In M — int(U), we define the 2- and 3— skeletons of
A’ to be those of A . However, we change the cell structure of int(U) as
follows: We add 2-cells D1, D2 to the 2-skeleton of A’ (note that they both
satisfy the twisting condition in Definition 1.6.1). Consider the 2-sphere
S 2 D1 U D2 where the union is taken along the common boundary G.
Let U’ be the 3-ball with 8U’ = S. Note that EIU' is tight as U’ C U and
5|U is tight. We add U’ and U — U’ to the 3-skeleton of A’ (note that
U — U’ can be considered as a 3-cell because observe that int(U — U’) is
homeomorphic to the interior of a 3—ball as in Figure 2.2(b)). Hence, we
established another contact cell decomposition of (M ,5 ) whose 1-skeleton
is G’ = G U J. (Equivalently, by Theorem 1.3.3, we are taking the connect
sum of (M, 5) with (S3,5st) along U’.) E]

25

2.1 The algorithm (The proof of Theorem 2.0.5)

Proof: By translating L in (R3,50) if necessary (without changing its
contact type), we can assume that the front projection of L onto the yz-
plane lying in the second quadrant { (y, z) I y < O, z > 0}. After an
appropriate Legendrian isotopy, we can assume that L consists of the line

segments contained in the lines

ki-

x 1,z=—y+a,-}, i=1,...,p,

II

II
r-‘H r-‘H

lj x=—1,z=y+bj},j=1,...,q

for some a1< a2 < < ap, 0 < b1< b2 < < bq, and also the line
segments (parallel to the x-axis) joining certain k,’s to certain lj’s. In
this representation, L seems to have corners. However, any corner of L
can be made smooth by a Legendrian isotopy changing only a very small
neighborhood of that corner.

Let it : R3 ———> R2 be the projection onto the yz-plane. Then we obtain
the square bridge diagram D = 7r(L) of L such that D consists of the line

segments

h,C7r(k,-) = {x=0,z=-—y+a,-}, i=1,...,p,

ij7r(lj) = {x=0,z=y+bj}, j=1,...,q.

26

Notice that D bounds a polygonal region P in the second quadrant of
the yz-plane, and divides it into finitely many subregions P1, . . . , Pm ( see
Figure 2.3-(a) ).

Throughout the proof, we will assume that the link L is not split (that is,
the region P has only one connected component). Such a restriction on L

will not affect the generality of our construction (see Remark 2.1.2).

 

ltd

 

 

 

 

a3 a4 a5

Figure 2.3. The region P for the right trefoil knot and its division.

Now we decompose P into finite number of ordered rectangular subregions
as follows: The collection {7r(lj) I j = 1,...,q} cuts each Pk into finitely
many rectangular regions RI,...,RIC"". Consider the set ‘3 of all such

rectangles in P. That is, we define

27

mé{RICIk/‘=17°Himl i=1,...,7nk}.

Clearly in decomposes P into rectangular regions ( see Figure 2.3-(b) ).
The boundary of an arbitrary element RIc in ill consists of four edges: Two
of them are the subsets of the lines raj-(kn), 7r(lj(k,,)+1), and the other two
are the subsets of the line segments hm“), hi2(k.l) where 1 S i1(k,l) <

i2(k,l) g p and IS j(k,l) < j(k,l) +1 3 9 (see Figure 2.4).

2 /\
bj(k,l)+1

7T(lj(k.l)+.1,l"""

hi2(kil)._. b
.‘ j(k,l)

 

h.
21 (N) ;::'7T(lj(k,l))

y

.3; \
/

amid) Clever)

Figure 2.4. An arbitrary element RI. in ‘3.

Since the region P has one connected component, the following holds for

the set ‘33:

(*) Any element of ‘13 has at least one common vertex with another element

of ‘43.

28

By (*), we can rename the elements of ‘13 by putting some order on them
so that any element of ‘3 has at least one vertex in common with the union
of all rectangles coming before itself with respect to the chosen order. More

precisely, we can write

‘I3={Rka=1,...,N}

(N is the total number of rectangles in ‘43) such that each R], has at least
one vertex in common with the union R1 U - - ' U Rk_1.

Equivalently, we can construct the polygonal region P by introducing the
building rectangles (Rk’s) one by one in the order given by the index set
{1,2, . . . , N}. In particular, this eliminates one of the indexes, i.e., we can
use Rk’s instead of RI’s. In Figure 2.5, how we build P is depicted for
the right trefoil knot (compare it with the previous picture given for P in
Figure 2.3-(b)).

Using the representation P = R1 U R2 U -- - U RN, we will construct the
contact cell decomposition (CCD) A. Consider the following infinite strips
which are parallel to the x-axis (they can be considered as the unions of

“small” contact planes along k,’s and lj ’3):

st = {1_egxg1+e, z=y+a,-}, i=1,...,p,

S7 = {—1—eSxS-—1+e, z=——y+bj},j=1,...,q.

29

 

Figure 2.5. The region P for the right trefoil knot.

Note that 7r(S,TI) = 7r(k,-) and 7r(SJ.,-’) = 7r(lj). Let R}, C P be given. Then
we can write

aRk=CIoC§UC£uC§

where GI C 7r(k,-1), GI C 7r(lj), G}: C 7r(k,-2), G: C 7r(lj+1) for some 1 3
i1 < i2 _<_ p and 1 Sj S 9. Lift CI,GI,G3,CI (along the x-axis) so that
the resulting lifts (which will be denoted by the same letters) are disjoint
Legendrian arcs contained in k,1,lJ-, kig, lj+1 and sitting on the corresponding

strips S-+ S.— 35,57

,1, J , 3+1. For l = 1,2,3,4, consider Legendrian linear arcs

II (parallel to the x-axis) running between the endpoints of CI’s as in
Figure 2.6-(a)&(b). Along each II the contact planes make a 90° left-twist.
Let BI be the narrow band obtained by following the contact planes along

II. Then define F1, to be the surface constructed by taking the union of

30

the compact subsets of the above strips (containing corresponding CI’s)
with the bands BI ’s (see Figure 2.6-(b)). GI’s and I I ’3 together build a

Legendrian unknot 7;, in (R3,50), i.e., we set
’71; =CIUIIUCIUIIUGIUIIUGIUII.

Note that 7r(7k) = BRk, ’yk sits on the surface Fk, and Fk deformation
retracts onto my. Indeed, by taking all strips and bands in the construction
small enough, we may assume that contact planes are tangent to the surface
Fk only along the core circle 7),. Thus, F], is the ribbon of ”WC. Observe
that, topologically, F1, is a positive (left-handed) Hopf band.

Let f}, : R}, ——> R3 be a function modelled by (a, b) +——> c = a2 — b2 (for an
appropriate choice of coordinates). The image fk(Rk) is, topologically, a
disk, and a compact subset of a saddle surface. Deform fk(Rk) to another
“saddle” disk D;c such that 8D,, 2 ’71: (see Figure 2.6-(c)). We observe
here that tw(ryk, Bk) 2 —1 because along 7k, contact planes rotate 90° in
the counter-clockwise direction exactly four times which makes one full left-
twist (enough to count the twists of the ribbon F}, since Fk rotates with
the contact planes along 7;, I).

We repeat the above process for each rectangle R}, in P and get the set
D: { Dk I DszkUik), k: 1,...,N}

consisting of the saddle disks. Note that by the construction of D, we have

the property:

31

 

 

(C)

Figure 2.6. (a) The Legendrian unknot 71,, (b) the ribbon Fk, (c) the disk
D], (shaded bands in (b) are the bands BI ’s).

(*) If any two elements of CD intersect each other, then they must inter-
sect along a contractible subset { a contractible union of linear arcs) of their
boundaries.

For instance, if the corresponding two rectangles (for two intersecting disks
in 53) have only one common vertex, then those disks intersect each other
along the (contractible) line segment parallel to the x-axis which is pro-

jected (by the map it) onto that vertex.

32

For each k, let DI be a disk constructed by perturbing Dk slightly by an

isotopy fixing only the boundary of Dk. Therefore, we have

(**) 80;. = 7,. = 31);, , int(Dk) n main) 2 i , and

tw(ryk, DI) = —1 = tw(yk, Dk).

In the following, we will define a sequence { Ak I k = 1, . . . , N } of CCD’s
for (S3,53t). A1,A2., and AI will denote the 1-skeleton, 2-skeleton, and
3—skeleton of A1,, respectively. First, take AI = '71, and A? = D1 U71 DI.
By (H), Al satisfies the conditions (1) and (2) of Definition 1.6.1. By
the construction, any pair of disks D1,,DI (together) bounds a Darboux
ball (tight 3—cell) U], in the tight manifold (R3,50). Therefore, if we take
A? = ’1 U5 (S3 — U1), we also achieve the condition (3) in Definition 1.6.1
(the boundary union “ U3” is taken along (9U1 = S2 = 6(S3 — U1)). Thus,
A1 is a CCD for (S3,53t).

Inductively, we define Ak from Ak_1 by setting

Al Ak—1U7k=’71U"'U'Yk—1U"/k.
AI = AI_1U 0,, U7], DI —_— D1U71D’1UMU Dk_1U,(,,_1)DI_, U Dk U.,,c DI,

A2 = U1U---UUk_1UUkUa(S3—U1U---UU;,_1UU;,.)

Actually, at each step of the induction, we are applying Lemma 2.0.6 to
Ak_1 to get Ak. We should make several remarks: First, by the construction

of yk’s, the set

(’71U°"U’Yk—1)rl”lk

33

is a contractible union of finitely many arcs. Therefore, the union AI_1U1;c
should be understood to be a set-theoretical union (not a topological gluing!)
which means that we are attaching only the (connected) part (yk\AI_1) of
7;, to construct the new l-skeleton AI. In terms of the language of Lemma
2.0.6, we are setting I = AI_1 \71, and J = ”We \ AI_1. Secondly, we have
to show that A2 = AI_1 U 0;, U7}, DI can be realized as the 2-skeleton of a
CCD: Inductively, we can achieve the twisting condition on 2-cells by using
(**). The fact that any two intersecting 2—cells in AI intersect each other
along some subset of the l-skeleton AI is guaranteed by the property (*) if
they have different index numbers, and guaranteed by (**) if they are of the
same index. Thirdly, we have to guarantee that 3-cells meet correctly: It is
clear that U1, . . . , U], meet with each other along subsets of the l-skeleton
AI(C AI). Observe that 8(U1Uo - oUUk) = S2 for any k = 1,. . .,N by (*)
and (ink). Therefore, we can always consider the complementary Darboux
ball S3 - U1 U U Uk_1 U Uk, and glue it to U1 U U Uk along their
common boundary 2-sphere. Hence, we have seen that A], is a CCD for
(53,530 with Legendrian 1-skeleton A1 = 71 U - ~ - U 7],.

To understand the ribbon, say 2),, of AI, observe that when we glue the
part 7;, \ AI_l of ”7/: to AI_1, actually we are attaching a 1-handle (whose
core interval is (7;, \ AI_1) \ 231,4) to the old ribbon 2k_1 (indeed, this
corresponds to a positive stabilization). We choose the 1-handle in such
a way that it also rotates with the contact planes. This is equivalent to
extending Zk_1 to a new surface by attaching the missing part (the part
which retracts onto (7;, \ AI_1) \ 21,4) of F], given in Figure 2.6-(c). The

34

new surface is the ribbon 2;, of the new 1-skeleton AI.

By taking k = N, we get a CCD AN of (S3,53t). By the construction, yk’s
are only piecewise smooth. We need a smooth embedding of L into the 1-
skeleton AI, (the union of all 'yk’s). Away from some small neighborhood
of the common corners of AI, and L (recall that L had corners before the
Legendrian isotopies), L is smoothly embedded in AIV. Around any com-
mon corner, we slightly perturb A}V using the isotopy used for smoothing
that corner of L. This guaranties the smooth Legendrian embedding of L
into the Legendrian graph AI, = UIV=1'yk. Similarly, any other corner in Alv
(which is not in L) can be made smooth using an appropriate Legendrian
isotopy.

As L is contained in the 1-skeleton AIV, L sits (as a smooth Legendrian
link) on the ribbon EN. Note that during the process we do not change
the contact type of L, so the contact (Thurston—Bennequin) framing of L
is still the same as what it was at the beginning. On the other hand, con-
sider tubular neighborhood N (L) of L in 2N. Being a subsurface of the
ribbon EN, N (L) is the ribbon of L. By definition, the contact framing of
any component of L is the one coming from the ribbon of that component.
Therefore, the contact framing and the N(L)-framing of L are the same.
Since N (L) C 2N, the framing which L gets from the ribbon EN is the
same as the contact framing of L. Finally, we observe that EN is a subsur-
face of the Seifert surface Firm of the torus link (or knot) an- To see this,
note that P is contained in the rectangular region, say PM, enclosed by the
lines 7r(k1),7r(kp),7r(l1),7r(lq). Divide PM into the rectangular subregions

35

using the lines 7r(k,-),rr(lj), i = 1,...,p, j = 1,...,q. Note that there are
exactly pg rectangles in the division. If we repeat the above process using
this division of PM, we get another CCD for (S3, 53,) with the ribbon FM.
Clearly, Fm contains our ribbon EN as a subsurface (indeed, there are
extra bands and parts of strips in Fm which are not in EN).

Thus, (1), (2) and (3) of the theorem are proved once we set A 2 AN,
(and so G = A1 , F = EN). To prove (4), recall that we are assuming

p > 3, q > 3. Then consider

k i total number of intersection points of all 7r(lj)’s with all his

That is, we define k i I{7r(lj) Ij= 1,...,q}fl {h,- I i = 1,...,p} I. Notice
that k is the number of bands used in the construction of the ribbon F,
and also that if D (so P) is not a single rectangle (equivalently p > 2,
q > 2), then k < pq. Since there are p + q disks in F, we compute the

Euler characteristic and genus of F as

2— — k (3F
x(F)=p+q—n=2—2g(F)—|aF|eg(p)=_§_fl+ I |

Similarly, there are p + q disks and pq bands in FM, so we get

2‘10—(1 Efl_Ian.ql

X(Fp.q)= p+q—pq = 2-29(Fp.q)—|3Fp,ql => 9(a))”: 2 2 2

Observe that IOFMI divides the greatest common divisor gcd(p, q) of p and

(1,80

langl S ng(Pa Q) S P 2} 9(Fp,q) Z +

2-p—q g
2 2

[\DI'B

36

Therefore, to conclude 9(F) < 9(Fp,q), it suffices to show that

pq-n>p-I3Fl-

To show the latter, we will show pq — k — p _>_ 0 (this will be enough since
I8F I 7E 0). Observe that pq — k is the number of bands (along x-axis) in
Fm which we omit to get the ribbon F. Therefore, we need to see that at
least p bands are omitted in the construction of F: The set of all bands

(along x-axis) in Fm corresponds to the set

era.) |j=1..-..q}fl{r(k.)li=1.~..p}-

Notice that while constructing F we omit at least 2 bands corre-
sponding to the intersections of the lines 7r(k1),7r(kp) with the family
{7r(lj) I j = 1,. . . ,q} (in some cases, one of these bands might correspond
to the intersection of the lines 7r(k2) or 7r(kp_1) with 7r(l1) or 7r(lq), but
the following argument still works because in such a case we can omit at
least 2 bands corresponding to two points on 7r(k2) or 7r(kp_1)). For the
remaining p — 2 line segments hg, . . . ,hp_1, there are two cases: Either
each h,, for i = 2,. . . , p —— 1 has at least one endpoint contained on a line
other than 7r(l1) or 7r(lq), or there exists a unique h,,1 < i < p, such that
its endpoints are on 7r(ll) and 7r(lq) (such an h,- must be unique since no
two vj’s are collinear I). If the first holds, then that endpoint corresponds
to the intersection of h,- with 7r(lJ-) for some j 79 1,9. Then the band

corresponding to either rr(ki) fl 7r(lj_1) or 7r(k,-) fl 7r(lj+1) is omitted in the

37

construction of F (recall how we divide P into rectangular regions). If
the second holds, then there is at least one line segment hi], which belongs
to the same component of L containing hi, such that we omit at least 2
points on 7r(k,v) (this is true again since no two v,- ’s are collinear). Hence,
in any case, we omit at least p bands from Fm to get F. This completes

the proof of Theorem 2.0.5. Cl

Corollary 2.1.1 Given L and Fm as in Theorem 2.0.5, there exists an

open book decomposition GB of (S3,5st) such that
1. L lies ( as a Legendrian link) on a page F of (28,
2. The page F is a subsurface of Fm
3. The page framing of L coming from F is equal to its contact framing,
4. pr > 3 and q > 3, then 9(F) is strictly less than 9(Fp3q), and

5. The monodromy h of GB is given by h = t71 o - - . o t.,N where 7;, is
the Legendrian unknot constructed in the proof of Theorem 2. 0.5, and

t7], denotes the positive {right—handed) Dehn twist along 7),.

Proof: The proofs of (1), (2), (3), and (4) immediately follow from
Theorem 2.0.5 and Lemma 1.6.3. To prove (5), observe that by adding
the missing part of each ’7}, to the previous 1-skeleton, and by extending
the previous ribbon by attaching the ribbon of the missing part of 7k

(which is topologically a 1-handle), we actually positively stabilize the

38

old ribbon with the positive Hopf band (H +, t7k). Therefore, (5) follows. El

With a little more care, sometimes we can decrease the number of 2-cells in

the final 2-skeleton. Also the algorithm can be modified for split links:

Remark 2.1.2 Under the notation used in the proof of Theorem 2. 0.5, we

have the following:

1. Suppose that the link L is split (30 P has at least two connected com-
ponents). Then we can modify the above algorithm 30 that Theorem

2. 0.5 still holds.

2. Let T]- denote the row ( or set) of rectangles ( or elements) in P ( or in
‘3) with bottom edges lying on the fixed line 7r(l3-) . Consider two consec-
utive rows Tj,TJ-+1 lying between the lines 7r(lj),7r(lj+1), and 7r(lj+2).
Let R E Tj and R’ C Tj+1 be two rectangles in P with boundaries

given as

8R=C1UC2UC3UC4, 6R’=C{UC§UC§UCI,

Suppose that R and R’ have one common boundary component lying
on 7r(lj+1), and two of the other components lie on the same lines
7r(k,-1),7r(k,-2) as in Figure 2. 7. Let 7,7’ C AI; and D,D’ C AN be the
corresponding Legendrian unknots and 2-cells of the CCD AN coming

from R, R’. That is,

8D 2 7, UD’ = 7’, and 7r(D) = R, 7r(D’) = R’

39

Suppose also that L 07 fl 7’ = (ll. Then in the construction of AN, we
can replace R, R’ C P with a single rectangle R” = R U R’. Equiva-
lently, we can take out 7 (17’ from AI, and replace D, D’ by a single
saddle disk D” with DD” = (7 U 7’) \ (7 fl 7’).

Z /\
j+2

,3 j+1

 

 

Figure 2.7. Replacing R, R’ with their union R”.

Proof: To prove each statement, we need to show that CCD structure
and all the conclusions in Theorem 2.0.5 are preserved after changing A N
the way described in the statement.

To prove (1), let P“), . . .,PI'") be the separate components of P. After
putting the corresponding separate components of L into appropriate po-

sitions (without changing their contact type) in (R3,50), we may assume

40

that the projection

P=PWUmUPW

of L onto the second quadrant of the yz-plane is given similar as the one
which we illustrated in Figure 2.8.

In such a projection, we require two important properties:

1. P“), . . . , PI’”) are located from left to right in the given order in the

region bounded by the lines 7r(k1), 7r(l1), and rr(lq).

2. Each of P0), . . . , PU”) has at least one edge on the line 7r(l1).

If the components P(1)...P(’”) remain separate, then our construction in
Theorem 2.0.5 cannot work (the complement of the union of 3-cells corre-
sponding to the rectangles in P would not be a Darboux ball; it would be
a genus m handle body). So we have to make sure that any component
P“) is connected to the some other via some bridge consisting of rectangles.
We choose only one rectangle for each bridge as follows: Let A; be the
rectangle in T1 (the row between 7r(ll) and rr(l2)) connecting P“) to Pa“)
for l = 1,. . . ,m — 1 (see Figure 2.8). Now, by adding 2- and 3-cells (cor-
responding to A1, . . . , Am_1), we can extend the CCD AN to get another
CCD for (S3, 53,). Therefore, we have modified our construction when L is
split.

To prove (2), if we replace D” in the way described above, then by
the construction of Air, we also replace two 3-cells with a single 3-cell
whose boundary is the union of D” and its isotopic copy. This alteration
of Afv does not change the fact that the boundary of the union of all

41

 

Figure 2.8. Modifying the algorithm for the case when L is split.

3—cells coming from all pairs of saddle disks is still homeomorphic to a
2—sphere S2, Therefore, we can still complete this union to S3 by gluing a
complementary Darboux ball. Thus, we still have a CCD. Note that 7 fl 7’
is taken away from the 1-skeleton. However, since L O 7 fl 7’ = 0, the new
1-skeleton still contains L. Observe also that this process does not change
the ribbon N (L) of L. Hence, the same conclusions in Theorem 2.0.5 are

satisfied by the new CCD. Cl

42

2.2 Examples

Example I. As the first example, let us finish the one which we have already
started in the previous section. Consider the Legendrian right trefoil knot
L (Figure 2.1) and the corresponding region P given in Figure 2.5. Then
we construct the 1-skeleton, the saddle disks, and the ribbon of the CCD
A as in Figure 2.10.

In Figure 2.10-(a), we show how to construct the l-skeleton G = A1 of A
starting from a single Legendrian are (labelled by the number “ 0 ”). We
add Legendrian arcs labelled by the pairs of numbers “1, 1” , . . . , “8, 8” to
the picture one by one (in this order). Each pair determines the endpoints
of the corresponding arc. These arcs represent the cores of the l-handles
building the page F (the ribbon of G ) of the corresponding open book 08 .
Note that by attaching each 1-handle, we (positively) stabilize the previous
ribbon by the positive Hopf band (H II , t7k) where ’71: is the boundary of the
saddle disk Dk as before. Therefore, the monodromy h of (98 supporting
(S3, €30 is given by

h=t710°°°0t78

where t7], E Aut(F,8F) denotes the positive (right-handed) Dehn twist
along 71,. To compute the genus 9p of F, observe that F is constructed

by attaching eight l-handles (bands) to a disk, and [SF I = 3 where [BF I is

43

the number of boundary components of F. Therefore,
x(F)=1—8=2—2gF—|8FI=>9F=3.

Now suppose that (Mft, 5%) is obtained by performing contact (in-surgery
on L. Clearly, the trefoil knot L sits as a Legendrian curve on F by our
construction, so by Theorem 1.5.1, we get the open book (F, hl) supporting

5 with monodromy

h, = t,1 o . . . o t,8 0t}:1 e Aut(F,6F).

Hence, we get an upper bound for the support genus invariant of 51, namely,

39(61) S 3 = 9F-

We note that the upper bound, which we can get for this particular case,
from [A0] and [St] is 6 where the page of the open book is the Seifert

surface F55 of the (5, 5)-torus link (see Figure 2.9).

44

 

 

Figure 2.9. Legendrian right trefoil knot sitting on F55.

45

Example II. Consider the Legendrian figure-eight knot L, and its square
bridge position given in Figure 2.11-(a) and (b). We get the corresponding
region P in Figure 2.11-(c). Using Remark 2.1.2 we replace R5 and R8 with
a single saddle disk. So this changes the set ‘3. Reindexing the rectangles in
‘33, we get the decomposition in Figure 2.12 which will be used to construct

the CCD A.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Figure 2.11. (a),(b) Legendrian figure-eight knot, (c) the region P.

In Figure 2.14—(a), similar to Example I, we construct the 1-skeleton G = A1
of A again by attaching Legendrian arcs (labelled by the pairs of numbers

“1, 1”,. . . , “10, 10”) to the initial are (labelled by the number “0”) in the

47

 

Figure 2.12. Modifying the region P.

given order. Again each pair determines the endpoints of the corresponding
arc, and the cores of the l-handles building the page F (of the corresponding
open book 03). Once again attaching each 1-handle is equivalent to (pos-
itively) stabilizing the previous ribbon by the positive Hopf band (H +, t7k)
for k = 1,. . . , 10. Therefore, the monodromy h of (93 supporting (33,530
is given by

hztvlo...ot710

To compute the genus 9;: of F, observe that F is constructed by attaching

ten l-handles (bands) to a disk, and IBF I = 5. Therefore,

x(F)=1—10=2—2gp—I5FI=>9F=3.

Let (M5, 53:) be a contact manifold obtained by performing contact (i1)-
surgery on the figure—8 knot L. Since L sits as a Legendrian curve on F

by our construction, Theorem 1.5.1 gives an open book (F, hg) supporting

48

52 with monodromy

122 = it,1 o - . . 0 15,10 o if e Aut(F, 8F).

Therefore, we get the upper bound 39(52) 3 3 = 9p. Once again we note
that the smallest possible upper bound, which we can get for this particular
case, using the method of [A0] and [St] is 10 where the page of the open

book is the Seifert surface F55 of the (6, 6)-torus link (see Figure 2.13).

 

All twists are left-handed

z—y=0

z+y=0

Figure 2.13. The figure-eight knot on Fee

49

......
and saddle d sssss

Chapter 3

Planar contact structures with

binding number three

In this chapter, we will obtain a complete list of contact manifolds cor-
responding to a fixed support genus and a fixed binding number. To get
such complete list, we consider all possible monodromy maps. Throughout
the chapter L(m, n) stands for the lens space obtained by —m/ n rational
surgery on an unknot. The first step in this direction is the following result
given in [E0].

Theorem 3.0.1 ([EOI) Suppose 5 is a contact structure on a 3-manifold

M that is supported by a planar open book (i.e., 39(5) 2 0). Then

1. If bn(5) = 1, then 5 is the standard tight contact structure on S3.

2. If bn(5) = 2 and 5 is tight, then 5 is the unique tight contact structure

on the lens space L(m,m — 1) r: L(m, —1) for some m E Z+ U {0}.

3. If bn(5) = 2 and 5 is overtwisted then 5 is the overtwisted con-
tact structure on L(m,1), for some m E Z+, with e(5) = 0 and
d3(5) = —;I-m + 3- where e(5) and d3(5) denotes the Euler class and

51

(lg—invariant of 5, respectively. When m is even then the refinement
of e(5) is given by I‘(5)(5) = g—l where 5 is the unique spin structure
on L(m,1) that extends over a two handle attached to a ,u with
framing zero. Here we are thinking of L(m, 1) as —m surgery on an

unknot and u is the meridian to the unknot. D

We remark that Theorem 3.0.1 gives the complete list of all contact 3-
manifolds which can be supported by planar open books whose pages have
at most 2 boundary components. Next step in this direction should be to
find all contact 3-manifolds (M, 5) such that 39(5) = 0 and bn(5) = 3. We
will not only get all such contact structures, but also distinguish tight ones
by looking at the monodromy maps of their corresponding open books (See
Theorem 3.0.2 and Theorem 3.0.4).

Let 2 be a planar compact oriented surface with I82] = 3. Consider the
boundary parallel curves a, b, c in E as in the Figure 3.1. Throughout the
chapter, 23 will always stand for this surface whose abstract picture is given
below. Consider Aut(2,82), the group of (isotopy classes of) diffeomor-
phisms of 2 which restrict to the identity on 82 (such diffeomorphisms are

automatically orientation-preserving).

It is known (see [Bi]) that

Aut(2,82) =z<oa>e z<ob)e 2(1),.) ӣ23

52

 

Figure 3.1. The surface 2 and the curves giving the generators of Aut(S, 82).

where D0, D5, Dc denote positive Dehn twists along the curves 0., b, 0 given
as in Figure 3.1. In the rest of the paper, we will not make any dis-
tinction between isotopy classes of arcs/curves/maps and the individual
arcs / curves / maps.

We start with studying the group Aut(E, 82) in details. Since generators

commute with each other, we have that

Aut(E, 823) = {Daprch'Ip,q,r E Z}.

For any given p, q, r E Z, let Y(p, q, r) denote the smooth 3-manifold given
by the smooth surgery diagram in Figure 3.2 (diagram on the left). It is
an easy exercise to check that Y(p, q,r) is indeed diffeomorphic to Seifert
fibered manifold given in Figure 3.2 (diagram on the right).

Now we state the following theorem characterizing all closed contact 3-

manifolds whose contact structures supported by open books (Dab =

DaPquDC") .

Theorem 3.0.2 Let (M, 5) be a contact manifold supported by the open

book (2,95) where (b = Daprchr E Aut(E,8E) for fixed integers p, q,r.

53

p+r q+r P

IIZ

(r in the box denotes the number of full twists)

Figure 3.2. Seifert fibered manifold Y(p, q, r).

Then (M, 5) is contactomorphic to (Y(p, q, r),5p,q,,.) where 5pm. is the con-
tact structure on Y(p, q, r) given by the contact surgery diagram in Figure

3.3. Moreover,
(I) 5 is tight (in fact holomorphically fillable) ifp 2 0,9 2 0, r 2 O, and

( 2) 5 is overtwisted otherwise.

  
 

lrl
copies

Figure 3.3. Contact manifold (Y(p, q, r),5p,q’,.).

54

Remark 3.0.3 In Figure 3.3, if r = 0, then we completely delete the family
corresponding to r from the diagram, 30 we are left with two families of
Legendrian curves which do not link to each other, and so the contact surgery
diagram gives a contact structure on the connected sum of two lens spaces.
However, if p = 0 (or q = 0), then we replace the Legendrian family
corresponding to p ( or g) by a single Legendrian unknot with tb number
equal to —1, and we do (+1)-contact surgery on the new unknot. Note
also that Figure 3.3 is symmetric with respect to p and g. This reduces the

number of cases in the proof of Theorem 3. 0.4.
Of course not all 5mm have binding number three. We will prove

Theorem 3.0.4 Let (M ,5) be a closed contact 3-manifold with 39(5) = 0
and bn(5) = 3. Then (M ,5) is contactomorphic to some (Y(p,q,r),5p,q,r)

satisfying the following conditions:
1. If r=0,thenp7é1andq;£1.
2. If r =1, then p E {—1,0} and q E {—1,0}.
3. If r=—1,thenp7é1andqaé1.
4-1fl7‘|_>_ 2, then re 74 -1 and (p,q) ¢ {(1.0),(0.1)}-

Suppose that (M, 5) is a closed contact 3-manifold with 39(5) = 0, bn(5) =
3, and let 01 = c1(5) E H 2(M ; Z) denote the first Chern class, and d3 =

d3(5) denote the 3-dimensional invariant (which lies in Q whenever c1 is

a torsion class in H 2(M ;Z)). Using 01 and d3, we can distinguish these

55

structures in most of the cases. In fact, we have either M is a lens space,
or a connected sum of lens spaces, or a Seifert fibered manifold with three
singular fibers. If one of the first two holds, then using the tables given
in Section 3.1, 3.2 and 3.3, one can get the complete list of all possible
(M ,5 ) without any repetition. That is, the contact structures in the list
are all distinct pairwise and unique up to isotopy. On the other hand, if the
third holds, we can also study them whenever c1 is a torsion class. More
discussion will be given later in Section 3.3.

We first prove that the submonoids Dehn+(2,82) and Veer(2,82) are

actually the same in our particular case.

Lemma 3.0.5 Dehn+(Z,8Z)= Veer(2,82) for the surface 2 given in

Figure 3.1.

Proof: The inclusion Dehn+(S, 8S) C Veer(S, 88) is true for a general
compact oriented surface S with boundary (see Lemma 2.5. in [?I for the
proof). Now, suppose that d) E Veer(2,82) C Aut(2,82) . Then we can

write gb in the form
45 = Daprch’" for some p, q,r E Z.

We will show that p 2 0,q _>_ 0,r _>_ 0. Consider the properly embedded
are or C 2 one of whose end points is x E 82 as shown in the Figure 3.4.
Note that, for any p, q, r E Z, Dc" fixes a, and also any image Daprq(a)
of (1 because c does not intersect any of these arcs. Assume at least one of
p, q, or r is strictly negative. First assume that p < 0. Then consider two
possible different images ¢(a) = Daprq(a) of a corresponding to whether

56

   
     

left circling

. right circling
I pI times

q times

left circling left circling
IqI times IpI times

p<0andq>0

p<0andq<0

Figure 3.4. The are a and its image ¢(oz) = Daprq(oz).

q < O or q > 0 (See Figure 3.4). Since we are not allowed to rotate any
boundary component, clearly ¢(a) is to the left of a at the boundary point
x. Equivalently, ¢(oz) is not to the right of a at x which implies that h is
not right-veering with respect to the boundary component parallel to a.
Therefore, qb E Veer(S, 82) which is a contradiction. Now by symmetry,
we are also done for the case 9 < 0. Finally, exactly the same argument
(with a different choice of are one of whose end points is on the bound-

ary component parallel to the curve c) will work for the case when r < 0. El

Lemma 3.0.6 Let (M ,5) be a contact manifold. Assume that 5 is sup-
ported by (Ed) where (b E Aut(2,82) . Then 5 is tight if and only if 5

is holomorpfically fillable.

Proof: Assume that 5 is tight. Since (I) E Aut(Z,8E) , there exists
integers p, q,r such that (b = DaPquDc’". As 5 is tight, the monodromy

57

of any open book supporting 5 is right-veering by Theorem 1.7.2. In
particular, we have 925 E Veer(E,8E) since (2,95) supports 5. Therefore,
gb E Dehn+(2,82) by Lemma 3.0.5, and so p 2 0,9 2 0,r 2 0. Thus,
5 is holomorphically fillable by Theorem 1.4.2. Converse statement is a

consequence of Theorem 1.4.1. I]

Now, the following corollary of Lemma 3.0.6 is immediate:

Corollary 3.0.7 Let (M ,5) be a contact manifold. Assume that 5 is sup-

ported by (2gb) where 95 E Aut(2,82) . Then

5 is tight 4:) (b = Daprchr with p 2 0,9 2 0,r Z 0. El

3.1 The proof of Theorem 3.0.2

Proof: Let (M ,5) be a contact manifold supported by the open book
(Egbmy) where am”. = Daprch’" E Aut(E,8E) for p, q,r E Z. As
explained in [E0], (M, 5) = (M(Ei¢p,q,r)’€(2i¢p.q.r)) is given by the contact
surgery diagram in Figure 3.5. Then we apply the algorithm given in [DC]
and [DGS] to convert each rational coefficient into :tl’s, and obtain the
diagram given in Figure 3.3.

To determine the topological (or smooth) type of (M ,5), we start with
the diagram in Figure 3.3. Then by converting the contact surgery coeffi-
cients into the smooth surgery coefficients, we get the corresponding smooth

surgery diagram in Figure 3.6 where each curve is an unknot.

58

l
l
’U
IIH
l-|

1

—1

b-Q

Figure 3.5. Contact surgery diagram corresponding to (2, 3pm.).

   

     

IqI — _q_ copies

P .
IpI — IF] copies [7'] copies IQI

Figure 3.6. Smooth surgery diagram corresponding to Figure 3.3.

Now we modify this diagram using a sequence of blow-ups and blow-downs.
These operations do not change smooth type of M. We first blow up the
diagram twice so that we unlink two —1 twists. Then we blow down each
unknot in the most left and the most right families. Finally we blow down
each unknot of the family in the middle. We illustrate these operations in
Figure 3.7. To keep track the surgery framings, we note that each blow-up

increases the framing of any unknot by 1 if the unknot passes through

59

 

_ _P_ - lql - -q- 0015ies
IPI [Pl comes IrI copies lql
p+r q+r
Y(p.q.r)

 

IrI copies

Figure 3.7. Sequence of blow-ups and blow-downs.

the corresponding twist box in Figure 3.6. So we get the first diagram in

Figure 3.7. Blowing each member down on the left (resp. right) decreases

_l
Iql"

Since there are IpI—— blow-downs on the left and IqI—— blow-downs

on the right, we get the second diagram in Figure 3.7. Finally, if we blow

the framing of the left (resp. right) +1-unknot by _lgl (resp.

down each (—fi)-unknot in the middle family, we getr the last diagram.

Note that each blow-down decreases the framing by— ,and introduces

a -|—| full twist. Hence, we showed that (M, 5) is contactomorphic to
r

(Y (1?, q, 7“,) 5qu,.) The statements (1) and (2) are the consequences of

60

Corollary 3.0.7. [I

We now examine the special case where Y(p, q, r) is homeomorphic to 3-
sphere S3. The following lemma lists all planar contact structures on S3

with binding number less than or equal to three.

Lemma 3.1.1 Suppose that (Y(p, q,r),5p,q,,.) is contactomorphic to (S3,5)
for some contact structure 5 on S3. Then Table 3.1 lists all possible val-
ues of (p, q, r), the corresponding 5 {in terms of the d3 -invariant), and its

binding number.

 

2" p q d3(€) bn(é)
........ T .3..... ......‘3..... ........1...... ......‘T1/.3.... ......3......
........ ‘72...“ .....f3.. .......1...... ‘1/3,3
........ T 1.... ......1.... ””2329... ......1/2... ......2...”
....... T 1 ........3...... -.....3/3..... ......3......
. .... 3 ............ 1. ............... 1. ...... T 112.13.131.13.) ......... 1. ......
3..... ........ : 1 ............. t 13/2 ..... ......3. ......
.IZ.._..3 ............. 1. .............. a .1 ............ 1. 12...... ......2 ......
......... 1. ......Q....... ........1.......7.143.111.1131).........1...._..
......... 1 -1...21.12..9..........1/.2............2......
......... 1‘3 ......f3...“ .....T.1/..3.... ......3...”
.........3...... ......3. .............. f .1... 3/23 ......
........3....... ......3 .............. 71... ......3/3.... ......3 ......
' IrI >2 1 —1 1/2 ' 2

 

 

 

 

Table 3.1. All planar contact structures on S3 with binding number 3 3.

Proof: The proof is the direct consequence of the discussion given in

the proof of Lemma 5.5 in [E0]. We remark that the interchanging p and

61

q does not affect the contact structure in Figure 3.3, so we do not list the
possibilities for (p, q, r) that differ by switching p and q. Note that in
Table 3.1 there are only two contact structures (up to isotopy) on S3 with

binding number 3, namely, the ones with d3-invariants —1/2 and 3/2. D

3.2 The proof of Theorem 3.0.4

Proof: We will use the results of Theorem 3.0.1, Theorem 3.0.2, and
Lemma 3.1.1. Consider the 3-sphere S3 in Theorem 3.0.1 as the lens space
L(l, 21:1). By Theorem 3.0.1, for any contact manifold (Y, n) with sg(77) = O

and bn(n) S 2, we have either
1' (K77) g (832631) 11137107) = 13
2. (Y, 77) 91 (L(m, —1),77m) for some m 2 2 if bn(n) = 2, and 77 is tight,

3. (Y, 77) ”E (L(m,1),77m) for some m 2 0 if bn(n) = 2, and n is over-

twisted (for m 71$ 0).

where 77m is the contact structure on the lens space L(m, —1) (or L(m, 1))
given by the contact surgery diagram consisting of a single family of Legen-
drian unknots (with Thurston-Benequen number —1) such that each mem-
ber links all the other members of the family once, and each contact surgery
coefficient is —1 (if nm is tight) or +1 (if nm is overtwisted). These are
illustrated by the diagrams (*) and (it) in Figure 3.8, respectively. Notice
the exceptional cases: m = 1 in (>2), and m = O in (7k).

62

5m+1

copies

m—1{

copies

 

(K271g (L(m -1),nm) (Y, n) g (L(m,1),17m)
77m a ways tlgflt, m 2 77m overtWISted if m _>_ 1
m = 1 => empty diagram 77m tlght If m = 0

Figure 3.8. Contact surgery diagrams for (Y, 77).

Now, if (M,5) is a contact manifold with sg(§) = 0 and bn(€) = 3, then by
the definitions of these invariants there exists an open book (2,4)) sup-
porting §. Therefore, by Theorem 3.0.2, (M,5) is contactomorphic to
(Y(p, q, r),€p,q,,.) for some p,q,r E Z, and the contact surgery diagram
of 5 is given in Figure 3.3. However, p,q,r can not be arbitrary integers
because there are several cases where the diagram in Figure 3.3 reduces to
either (>2) or (*) in Figure 3.8 for some m. So for those values of p, q,r,
(M,§) can not be contactomorphic to (Y(p,q,r),€p,q,,.) 2’ (Y, 17) because
bn(5) = 3 # 2 _>_ bn(n). Therefore, we have to determine those cases.

If [p] 2 2 and |q| 2 2, then the only triples (p, q,r) giving L(m,:l:1)’s
are (—2,q,1) and (2,q, —1). Furthermore, if we assume also that |r| > 1,
then the Seifert fibered manifolds Y(p, q, r) are not homeomorphic to even
a lens space L(m, n) for any m,n (for instance, see Chapter 5 in [Or]).
As a result, we immediately obtain bn(fp,q,r) = 3 for |p| _>_ 2 and |q| 2 2

and Ir] 2 2. Therefore, to finish the proof of the theorem, it is enough to

63

analyze the cases where |p| < 2 or |q| < 2, and the cases (—2, q, 1) and
(2,q, —1) for any q. As we remarked before, we do not need to list the
possibilities for (p, q,r) that differ by switching p and q. We first consider
r = 0, i1, :l:2, and then the cases r > 2 and r < —2. In Table 3.2 - 3.8,

we list all possible (M, .f) for each of these cases.

Remark 3.2.1 T 0 determine the binding number bn({) in any row of any
table below, we simply first check the topological type of the manifold under
consideration. If M m S3, we determine the corresponding binding number
using Table 3.1. If the topological type is not L(m, 1) or L(m, —1), then
we immediately get that bn(é) = 3. If M x L(m, 1) with m > 1, then we
first compute c1(£). If c1(€) 7g 0, then bn(5) = 3 as 01(77m) = 0 for any 77m
given above. If 01(5) :0, we compute the d3(§) using the 4-manifold defined
by the surgery diagram in Figure 3. 6. (Indeed, we can use the formula for d3
given in Corollary 1.8.2 as long as c1(§) is torsion. In particular, whenever
H2(M) is finite, then d3 is computable). Then if d3(£) = d3(77m) = (—m +
3) /4, then g is isotopic to nm which implies that bn(é) = 2 by Theorem
3.0.1. Otherwise bn(5) = 3. In the case that M 2 L(m, —1) with m > 1,
we first ask if E is tight. If it is tight (which is the case if and only if
p Z 0,q 2 0,r 2 0), then bn(§) = 2 (again by Theorem 3.0.1) since the
tight structure on L(m, —1) is unique (upto isotopy). If it is overtwisted
(which is the case if and only if at least one of p,q,r is negative), then
bn(é) = 3 because 5 is not covered in Theorem 3.0.1. As a final remark,

sometimes the contact structure 5 can be viewed as a positive stabilization

64

of some nm. For these cases we immediately obtain that bn(5) = 2 because

positive stabilizations do not change the isotopy classes of contact structures.

To compute the d3-invariant of {p,q,r (for c1(§p‘q,,.) torsion), we will use the
(n+ 1) X (n+ 1) matrices An (n 21), 13,, (n 2 1), and Cu (n 2 4) given

below. It is a standard exercise to check that
1. 0(An)=n—1ifn21,and 0(Cn)=n—1ifn24.
2. 0(Bn) =n—3 if n 2 3, and 0(Bn) =0 if n: 1,2.

3. The system An[b]£+1 = [0]:+1 has trivial solution [b]3,3+1 = [O]:+1
where

[b]n+1 2 [b1 b2 - - - bn+1], [0]n+1 = [O 0- - - 0] are (n+1) x 1 row matrices.

f.0—1—1...--F (o-1—1...—U Fo-1-1...-17
—10—1...—1 —10—1...—1 —10—1...—1
An: _1—1.. : Bn: —1_1-. : Cu: _1_1- o
0-1 - 0—1 22 0—1
t1—1""10J E1—1---—1—g C1—1""13J

 

 

 

 

 

 

In some cases, An appears (as a block matrix) in the linking matrix CM”.
of the framed link ILMJ. given in Figure 3.6. On the other hand, Bn and
0,, are very handy when we diagonalize LIN”. to find its signature. As we

discussed before, the link lbw”. defines a 4-manifold anr with 8X = M.

So we have

65

0(Xp.q,r) : U(£p,q,r)a
X(Xp,q,r) = 1 + (# of components of filmy),
62 = 1b] k£p,q,r [1311

where [b]{ is the solution to the linear system
Ep,q,r[b]7kr = [rot(Kl) T0t(K2) - --rot(Kk)]T

with K1, K2, - - - K k being the components of 1112,92"

To compute the first Chern class 01 (€12.92) 6 H 2(M ), note that in Figure
3.3, the rotation number of any member in the family corresponding to r is
i1 (depending on how we orient them). We will always orient them so that
their rotation numbers are all +1. On the other hand, the rotation number
is O for any member in the family corresponding to p and q. Therefore,
c1(€p,q,r) = PD‘1(u1 + [1.2 + - ° - + #Irll where u,- E H1(M) is the class of the
meridian of the Legendrian knot K ,- in the family corresponding to r. Then
we compute H1(M) (which is isomorphic to H 2(M ) by Poincaré duality)

as

H1(M) = (flu/12"" ,Mkl Cami/111 = [011)

where [,u];c = [m Mn- [11,] is the k x 1 row matrix. The final step is to
understand PD(c1 (5194,») = 111 + 112 + + “Irl in this presentation of
H1(M).

In Table 3.2, we need to compute the binding number bn(5) for the rows
5, 12. For the other rows, see Remark 3.2.1.

66

 

 

 

 

 

 

 

 

r p q resultingM bn(5) diagram for§c1(§)EH2(M) d3(€)

.9. :1 ...-.11 ............ .53-... ..3... ...131131119.3.... .....11..§.{11.}...... .....3/3 .....
..0.. :1 ............. 3393.3.1..>.<..S..3.....,..2 ...... 1:“ 1.2.3.322... .....9..€..3 ............. 1 .......
..0. 7.1....1 ............. 333.131.771.31... ....9..€..{0.}. .......... 1 1.2. .....
.9. 71.9.2.3. ..333319::1).... ...2 ...... F 12.11323 ........... [0.] ........... ( .9.fr..1)/.1.
.9. :1 11.3.73. ..533911.‘1121.1.... ..3 ...... 13 15.1.1123 ........... [.01 ......... (7.11117571/4
2.3. .0. ........ t??3.3"’..-.......3 ...... 3? 3.33.333 ...... [31.22.32 ....... 1. /2.....
.9. .9 1. ....... 31.232323...” .2. ..1fil1’1f11. 062 0
9..? 9.2.2..313.333.13.191i1h2 ...... F 1.2.3.1322 ........... [0.] ........... ( 9..-.12/1
..o. ..3 33:2. 31.322.311.119la1). 3.. ...3323333 ........... [9.1 ......... <..-.I.ql.1.3>/4
.22. 1. ............ .3? ............ 1.1....(3.>..3.3:.......9.e..{9.}. ......... : 1/2...
2.1. 9.2.2....33331951.) ..... l.._2..._.<:1.>.1339 .......... [ 9.1 ........... ( 3..-.322.
0 lqs-z S3#L(Iql,1) 12 (*)m=lql [0] (—lql+3)/4

Table3.2. The case r=0 (IpI < 2 or |q| <2).

0 pr = —1,q S —2, r = 0, we need to compute d3(§_1,q,0) as c1(§_1,q,0) = 0:

We have

A1 0

£_17q30 :
0 Alql+l
The contact structure 6-1,“), and 114-1,“) descrlbmg X_1,q,0 are glven 1n

Figure 3.9. We compute that s = lql + 3, c2 = 0, x(X_1,q,0) = Iql + 4, and

0(X—1,q.o) = 0(A1) + 0(Aqu) = 0 +IqI—1=IqI—1.

and so we obtain d3(§_1,q,0) = (-|q| + 7) /4 by Corollary 1.8.2. Therefore,
$-14”) is not isotopic to 77l9| as d3(77|q|) = (—|q|+3)/4. Hence, bn(€_1,q,0) = 3
for any q S —2 by Theorem 3.0.1.

o If p = 1, q 3 —2, r = 0, we have (2, 931.20) = S;(H+, Dz) (recall the iden-

tification of E and the curves a,b, c in Figure 3.1). Therefore, 5mg '5 77qu

67

 

3}|ql+1(b:‘lilliiib
1copies

Figure 3.9. (a) €_1,q,0 on S3#L(|q|, 1) z L(|q|,1), (b) the link 114-1,”).

since (H +,Dg) supports the overtwisted structure 77l9| on L(|q|.1). Hence,

bn(élgp) == 2 for q S —2.

In Table 3.3, we need to compute the binding number bn(E) for the rows
1 and 9. For the other rows, see Remark 3.2.1.
o If p = —2,q S —4,r = 1, let Ki’s be the components (with the given

orientations) of ll._2,q,1 as in Figure 3.10. Then we obtain the linking matrix

(—3 —1 —1 —1 —1 —1)
—1 0 —1 —1 0 0
—1 —1 O —1 0 O
—1 —1 —1 0 0 O
£—2,q,1= —1 0 O 0
Alql

 

 

68

 

7‘ P 9 resultingM 1711(5) 61(€)€H3(M) (13(6)

1 —2qS-4 L(|9+2l,1) 3 [|q|—4] (-q2—7q—14)/(—4q—8)
..1..-2...:3 ............. 33311122331313: if:if€il2Iffffifi
..1. .T3 ...T.3 .......... 33.3.33....... ...3 ....... T 3.9.3. .................... 9.1 .9 ...............
1 -2 922 L(9+2,-1) 3 [q] (q3+q+2)/(4q+8)
..1. 71.211211 ........... 53W...2:.II331{2}I:IIiIII:IiiiliI/éiiffff.
..1. .11....9 ........... 33X533 053 ...............3 ................
..1. .0 ....1 .............. :33 .............. 1 .. .....9.€..{111 .................. T. 1/3. ...............

..1. .0 .933 ....... 3111.17.11.........3 ........... l .0] .................... 1 9.13114 ...........

..1. .0 9.5.11 ...... 3119.1.3.1l........ ...3 ........... [Q] .................. ( .T.|.2|.J.r..3.)/.’1 .........

..1. .1 ...T.3 ......... 9.1.3.1711“..... ..3. ........... l 1.] ........................ 1 ./.3. ..............

..1. .1 ...1 ........... 31311.)............3 ........... l .1] ....................... T .1/3 .............
1 1 43—3 L(2q+1,_q_1) 3 [|q|-ll (-93-49-2)/(-49-2)
1 1 922 L(29+1,-9-1) 3 [9+1] (93-29-1)/(49+2)

 

 

 

 

 

 

Table 3.3. The case r = 1, |p| < 2 or |q| < 2 (and the case (p,q,r) =
("—23% 1))

It is not hard to see that

H1(M) : (#1) #2) ' 3 ' aulql+51 £—2,q,1[/’l’]|j;|+5 = [DIEM—5 >

= (ml (|q| - 2)/22 = 0 > ’5 Zia—2.

and ,ul 2 (|q| — 4)u2. Therefore,

CI(€—2,q,1) = P171011) = 1313—36311 _ 4M2 = “(II _ 4] E ZIqI—2-

Thus, if q < —4, then 54.9.1 is not isotopic to 77mm as c1(nlq+2|) = 0
implying that bn(€_2,q,1) = 3 by Theorem 3.0.1. If q = —4, we compute
that d3(€_2,_4,1) = —1/4 sé 1/4 = d3(772), so bn(§_2,_4,1) = 3.

o If p = 0,q _<_ —-1,r = 1, we have (E,¢0,q,1) = S:(H+,Dg) (again re-

call the identification of Z and the curves a, b,c in Figure 3.1). There-

69

 

 

 

Figure 3.10. (a) 5-2,,“ on L(|q + 2|,1), (b) the link 14-2,“.

fore, 50,9,1 '5 77qu since (H +, D3) supports the overtwisted structure 17qu on

L(|q|,1). Hence, bn(fom) = 2 for q < O.

In Table 3.4, we need to determine the binding number bn(fi) for the rows
4, 7, 9, and 11. For the other rows, see Remark 3.2.1.
o If p = 2,q S —2,r = —1, then using the corresponding matrix £2,q,_1,

we have

H1(M) = (111,112,"' ,Mlql+3l £2,q,—l[#l|7(;|+3 = 1Ol£|+3l

= (1121 (191+ 3)#2 = 0) 2' Z|q|+2,

and u] = |q|u2. Therefore,

01(52,q,—1) = 1D13—3011) = 1313—3691112): ”Q” E Z|q|+2-

Thus, if q s —2, then {2471 is not isotopic to "la—2| as 01(17|q—2|) = 0

implying that bn(§2,q,_1) = 3 by Theorem 3.0.1.

70

 

r p q resultingM bn(E) 01(6) 6 H2(M) d3(£)

—-1 2 924 L(q-2,—1) 3 [q-4] (—q2+3q—6)/(—4q+8)
7.1.2....2 .............. 32....flflj [3: fi.11if€:.{'11.}:.f3.:1:12:31211311:if
.T.1.3.....3 ........... 33.3.33....... ...3 ...... T 3.6.3 ..................... 3. .9 ................
—1 2 (q s —2 L(|q—2M) 3 [|q|] (---q2 -3q+6)/(—4q+8)
:1 ..1. 21.1.2.1 ........... Sim... 1.2...1:119:111111:311.111::11211211
.T.1..Q. ....11 ........... 33.3.33....... ..3... ”H.393. ...................... 1 ..................
.T.1..11. ...T.1 ............. 33........... ..3... ....9.‘.5..{111...................3../.3 .................
.T.1.9...‘1..?.1 ....... 13.1113.f.1).........3 .......... l 9.] .................... ( {1.15.1114 .............
.T.1..Q. 9.5.11 ...... 15119.1.3.1.)........ ...3. .......... l 9.] ................. ( ,TI,‘Il.'lT.fl./.3 ...........
.T.1.T.1...T.1.-... ..... .1113C.1l....... ..3. .......... l 1..-.......l............‘1./.3 .................
.T.1.T1.....3 ..... l ...... 91311.)......... ...3. .......... l 1....l..... ...........3/.3. .................
-1 -lq S —2 L(-39+1,-9+1) 3 l|9|+ll (-92 -69+3)/(-49+2)
—1 —1 q23 L(-29+1,-9+1) 3 19-1] (-93)/(-49+2)

 

 

 

 

 

 

Table 3.4. The case r = —1, |p| < 2 or |q| < 2 (and the case (p,q,r) =
(23q9—1))'

o If p = 0,q S —1,r = —1 (the rows 7 or 9), then c1(§o,q,_1)= O and so

we need to compute d3(£0,q,_1). Let Ki’s be the components of 140%-] as

in Figure 3.11. Then

£0’q,_1 2

(—1—1—1

 

 

71

(1 0)

0—100

00

 

 

By diagonalizing the first two rows of £0,q,_1, we obtain the matrix on the
right. So o(£o,q,_1) = 0(AIQI) = |q| — 1. The contact surgery diagram for
€0,q,_1 and the corresponding 4-manifold X0,q,_1 (with 6X0,q,_1 = M) are

given in Figure 3.11.

 

 

 

Figure 3.11. (a) €0,654 on L(|q|,1), (b) the link 140%-].

Then the system £0,q,_1[b]T = [rot(Kl) rot(K2)---rot(Klql+3)]T =
[10 0~-0]T has the solution [b] = [0 —1 0m0], and so c2 = 0.
Moreover, x(X0,q,_1) = Iql + 4 and s = |q| + 3. Therefore, we obtain
d3(§o,q,_1) = (—|q| + 7)/4 implying that 50.9.4 is not isotopic to 77qu as
d3(r}lql) = (—|q| + 3)/4. Hence, bn(€0,q,_1) = 3 by Theorem 3.0.1.

o If p = ——1,q = 2,r = ——1, we have c1(§_1,2,_1) = [1] implying that
bn(§_1,2,_1) = 3. To see this, note that c1(§—1,2,—1) = [DD—301,1) where

111 is the meridian of the surgery curve corresponding K1. Then using

72

W6 get H1(M) = ( #19u2,#3,u4| 13—12—de = [0]

£—1,2,—1 =

 

—1 —2 0 0
—1 0 0 ——1
(—1 0 —1 0)

Z3, and ,ul 2 —2u2. Therefore, we compute

 

T

4l:(/12l3112=0>g

01(9—1,2,—1) = P1731011) = PD_1(T3112) = [T3] 6 Z3 5 l1] 6 Z3.

 

 

 

 

 

 

 

r p q resultingM bn(éf) cl(€)EH2(M) d3(§)

2 —1 q: —3 L(|q—211) 3 [2] (—q2—3q+6)/(—4q+8)
f2£1]9391fiffi191<3£1fi>fiiiijiiII31.ifffili2lffiii iIiiZilf}.3i/.3iif...ifl. ....
..3.T.1.‘1..‘.-‘.T.3 ........ Pif13.1.).........._...3 ............ l3] ...................... T .1./.‘.1 ...............
..3 .T1....1. .............. .33 .................. 3 ........9.€..{11.}. ................... 1 ./.3 ................
..3. .T1..3 ........... 53XS333€3 ..................... 9.3 .9 .................
..3 .T1....3. .............. 5330€{0} 3/3 .................

2 —1 q24 . L(9—2,—1) 3 [q—4] (q2—3q+6)/(4q—8)
.23I19:2:113:13321112;}.11II3CIIIIIIldlIIiIZQiii....i..I..1./.4. ................
..3..0. ...... 1 ......... 3. 3#1.(3z.T.1l..........3 ............ [01 .................... T .1/.‘.1 ................
..3 .9. .931..3123.T.1113.Ii(.32f.11....3 ............ [01 .................. ( .11..T.3.)./.‘.1 .............
..3 .9. ..‘1..€.9..399.12.11.131132T1)“....3. ............ [01 ................ ( _.-.|.9|.t.‘1)./.‘1 ...........
..3. ..1 ...T3 .......... 313’.f.11............3 ............ l3] ....................... 1 ./.3. ................
.31. ....1 ........... 111T.5.2.31.............3 ............ [.31 .................... T .1/111. ...........

2 1 (13—3 L(-3q-2.q+1) 3 [2] (3q2+15q+10)/(12q+8)

2 1 q22 L(-39-2,q+1) 3 [2] (3q2—3q—2)/(12q+8)

Table3.5. The case r=2 (|p| < 2 or |q| <2).

73

In Table 3.5, we need to compute the binding number bn(é) for the rows
1, 2, and 3. For the other rows, again see Remark 3.2.1.

For the first three rows in Table 3.5, the contact structure 5-1413 on L(|q —
2|, 1) and the link ll._1,q,2 (q _<_ —1) are given in Figure 3.12. We write the
linking matrix £_1,q,2 as the matrix on the left below. It is not hard to see
that c1(€_1,q,2) = [2] E Zlq-2|3 and so bn(£_1,q,2) = 3. As an illustration we
will compute d3(€_1,q,2) (even though it is not necessary for the proof). The
matrix on the right below is obtained by diagonalizing the first two rows of
£_1,q,2. So we compute o(£_1,q,2) = 2 + 0(A1) + ”(quI) which is |q| — 1 if
q _<_ ——3, and is equal to 2 if q = —1,—2 (recall 0(Bn) is n — 3 if n 2 3,

and 0 if n = 1, 2). By a standard calculation, the system

(13—2—1—r—1—1 —1 ’2 ago 0; 0 0- - a
—2—3 —1—1 —1—1 —1 01/2 0 0; 0 0 0
:Ilr .......... innuomuumo .010? .......... 51020 .......... 0
—4—422412 0 0--- 0 10 023415 0 0- - 0
£—1q2 : —1—1300~ ..................... _’ .0...O.:...O.....0.: .....................
” —1—1590- A 90390- B
3 - |q| ; ' - |q|
(:1—1 0 0 J (9 0 0 0 J

 

 

 

 

£_1,q,2[b]T = [rot(Kl) T0t(K2) - - ~rot(K|ql+5)]T = [1 1 0- - - 0]T

has the solution

 

 

|9| |9| -2M| -2M| -2 -2 1

b:
11 [lql+2|9|+2|q|+2|q|+2|q|+2 |q|+2

74

for q S —1, and so we compute

C3 = [blE—1,q,2[blT = 2|9I/(I9I + 2)-

 
   
 

 

 

 

 

 

 

 

/
h /\ I <5
eac /
- C / . 1
frammgk/ ,[1‘11 ‘1' 0

+ 1 copies

      

Figure 3.12. (a) 5-1"]; on L(|q — 2|, 1) for q S —1, (b) the link IL_1,q,2.

o If p = —1,q = —1,r = 2, then c2 = 2/3, 0(X_1,_1,2) = 2, x(X_1,_1,2) =
7, and s = 4. So we get d3(§_1,_1,2) = —5/6.

a If p = —1,q = —2,r = 2, then c2 =1, 0(X_1,_2,2)= 2, x(X-1,_2,2) = 8,
and s = 5. Therefore, we get d3(€_1,_2,2) = —1/4.

- 13p = —1,q s —3,r = 2, then c3 = 2|ql/(lql + 2), 0021.12): |q|—1,

x(X_1,q,2) = |q| + 6, and s = |q| + 3. So we obtain

2
—q -— 3g + 6
d3(€-1,Qa2) : _4q + 8 '

 

75

 

 

 

 

 

 

 

r p q resultingM bn(€)cl(€)€H2(M) d3(§)

—21 c123 L(9+2,—1) 3 [2] (92+q+2)/(4q+8)
:2...1.......2 ......... .111.:1>..........3.. ......iiliilfifiiiifiIff.iffii1l2.fi.fiiI:
.T.3...1 ....... 1 ......... .L. 13.3.i11......,..3 .......... [.3.] .......................... 1 [.3 ...............
.T.3...1 ..... T .1 ............ 33..............3..,....9..€..{9.}. ...................... 1/3. ...............
.T.3...1 ..... T .3 ......... 33.3.33....... ....3. ......3..€..3 ........................ 3. .9 ...............
.T.3...1 ..... T .3 ............ 33W ...3.. ....9..€..{9.}. ..................... T.1/.3 ..............
—2 1 qS-4 L(|9+2|,1) 3 [|q|-4] (-93-79-14)/(-49-8)
:2..9.....I.0f]21.23.9112,:12}:i.f.iildl..fii1:91.11:3132111111:
.T.3....0. ....... 1 ........ 3 3.33113211.....3 ......... [Ql ......................... 1 (.4. ...............
.T3...0. ..... T .1 ........ 33#.L.(3z1)......3 .......... lQl...................”...3/F4. ...............
:.-.2..9...9.2_2..1191:112112211...3. .......... [019/4 ...............
:2..11.9§..-_2 1119.|z.1.>#1.<.22.1). ...3. .......... [01 .................. ( ..—.I.9I.+. 31/1 ...........
.T.3..T.1......3 ......... 151311.)............3 .......... 1.0.] .......................... 1. [.3 ................
.T.3..T.1...T1 ........ 917.52.731.22. ..3 .......... [.3.] .......... 1 ............. 1 1X19. ..............
—2 —1qg—2 L(3q-2,q—1) 3 [2] (—3q2—15q+10)/(—12q+8)
—2 —1|9|23 1439-29-1) 3 [2] (—3q2+3q—2)/(—12q+8)

Table 3.6. The case r = —2 (lpl < 2 or |q| < 2).

In Table 3.6, we need to compute the binding number bn(é) for the rows
7, 9, 10, and 13. For the other rows, see Remark 3.2.1.

o If p = 1,q S —4,r = ——2, the contact structure €1,q,_2 on L(|q + 2|, 1)
and the link L1,q,_2 are given in Figure 3.13.

We will first compute that c1(€1,q,_2) = [|q| —4] E ZIql—2 (so bn(§1,q,_2) = 3),
and then (even though it is not necessary for the proof) we will evaluate

d3(€1,q,_2) as an another sample computation. Using £1,q,_2 (on the left

76

 

 
  
  

 

 

 

 

 

each
fram1ng4\/> “(II + 1
, copies
K

|¢1l+3

 

Figure 3.13. (a) {mfg on L(|q + 2|, 1) for q < —3, (b) the link L1,q,_2.

below), we have

= (HuMl _ 3#1 T (M +1)#3 = 0, T2IJ1 T |€1|H3 = 0)
= < #31 (ICII _ 2M3 = 0) '5 Zqu—m
and also we have ,ul = #2 = —p3. Therefore, we obtain

Cl(€2,q,—1) = I’D-1011+ #2) = PD—1(T2M3) = ..2 5 “(1| T 4] E Z|q|—2-

The matrix on the right below is obtained by diagonalizing the first two
rows of £1,q,_2. So we compute 0(£1,q,_2) = 0 + 0(C’lql) 2 lg] — I (recall
0(Cn)=n—1ifn22).

By a standard calculation, the system

£1,q,_2[b]T = [rot(Kl) rot(Kg) - ~ - rot(Klql+3)]T = [1 1 O - - -O]T

77

 

 

 

 

—2_1;_1.. -1 0-1/220 ---0
£1q_2 = —'I'_'.'E """"""""" _) Dun-0...; ...............
. AIQI I ' 0qu
-ICI| -Iq| 2 2

], and so we obtain

 

has the solution [b] = [lql __ 2 |q| _ 2 |q| _ 2 |q| _ 2

c2 -—- [bl£1,q,—2[blT = -2lq|/(|q| — 2).

Moreover, x(X1,q,_2) = |q| + 4, and 3 = |q| + 3. So we compute

2
—q — 7q — 14
d3(€1,q,_2) 2 —4q _ 8 .

 

o If p = 0,q = 1,7" = —2, then {0,1,4 and Lula—2 are given in Figure 3.14.

(b)

 

 

 

 

 

 

 

Figure 3.14. (a) €0,1,_2 Ol'l S3#L(2,1) % L(2,1), (b) the link 1140,1,_2.

78

One can get C1(€o,1,—2) = 0, so we need d3(§0,1,_2). The corresponding

linking matrix is

(_1—2_1\ L100)

£0,1,_2= —2 —1—1 —+ 0 2 0
(4-10) (001)

We diagonalize £0,1,-2, and obtain the matrix on the right. So 0(£0,1,—2) =

 

 

 

 

1. We find that the system £0,1,_2[b]T = [rot(Kl) 7'0t(K2) rot(K3)]T =
[1 1 0]T has the solution [b] = [0 O —— 1], and so c2 = 0. Also we have
x(X0,1,_2) = 4 and s = 3. So we get d3(§0,1,_2) = 1/4 = (13(772) which
implies that €0,1,_2 is isotopic to 772. Thus, bn({0,1,_2) = 2 by Theorem
3.0.1.

o If p = 0,q = —1,7‘ = —2, then the contact structure {0,172 on L(2, 1)
and the link 1407172 describing X0,_1,_2 are given in Figure 3.15. It is easy
to check 01 (€0,_1,_2) = 0, so we compute d3(£0,_1,_2):

The corresponding linking matrix is

£0,-1,_2= —1—1 0 0 0 ——> 0 010 0

 

 

 

 

79

 

 

 

 

 

 

 

Figure 3.15. (a) {0,4,4 on S3#L(2, 1) z L(2,1), (b) the link L0,_1,_2.

We diagonalize £0,472, and obtain the matrix on the right. So

0(£0,_1,_2) = 1. The system
£0,_1,_2[b]T = [rot(Kl) T0t(K2) rot(K3) rot(K4) 7"0t(K5)]T = [1 1 O O 0]T

has the solution [b] = [0 0 —— 1 O O] which yields 02 = 0. Also we have
X(X0,—1,—2) = 4 and 3 = 3- so we get d3(€0,—1,—2) = 5/4 75 1/4 = (13(02)-
Therefore, {071,4 is not isotopic to 772, and so bn(fo,_1,_2) = 3 by Theorem
3.0.1.

o If p = —1,q = 2,r = —2, then the contact structure §_1,2,_2 on L(4, 1)
and the link 144,272 describing X_1,2,_2 are given in Figure 3.16. We

compute that cl(§_1,2,_2) = 0, so we need to find d3(§_1,2,_2).

80

 

 

 

 

 

 

 

 

Figure 3.16. (a) 6-13:2 on L(4,1), (b) the link L_1,2,_2.

The corresponding linking matrix is

 

 

 

 

(—1—2—1—1—1\ (2 0 000)
—2—1—1—1—1 0—1000
£_1,2,_2= —1—1—2 0 0 —> 0 0 —100
—1—10 0—1 0 0 010
[—1—10—1 0) \0 0 002)

We diagonalize £_1,2,_2, and obtain the matrix on the right. So

a(£_1,2,_2) = 1. The system
£_1,2,_2[b]T = [rot(Kl) rot(Kg) rot(K3) rot(K4) 'rot(K5)]T = [1 1 O 0 0]T

has the solution [b] = [1/2 1/2 — 1/2 — 1 — 1], so we compute
c2 = 1. Moreover, x(X_1,2,_2) = 6 and s = 4. Then we get d3(€_1,2,_2) =

1/2 aé —1/4 = d3(774). Therefore, §_1,2,_2 is not isotopic to 774, and so

81

bn(§_1,2,_2) = 3 by Theorem 3.0.1.

 

P q 01(6) 6 H2(M) (13(5)
.T.1. ....Q ............ [.01 ................................... ( I .T.1l/f1 ...........................
:1 ..... 1. ....Q.€..{0.}. .................................. 1 _/.2. ...............................
.T.1. ....2. ............. [.21 ........................... ( T113? T.§l/.(.‘1T_.T.3) ...................
—1 qS --1 [Tl (92T+qr2~q2-r2-q-rl/(4qr-4q-4T)
—1 q23 [r] (q2r+qr2—q2—r2——6qr+5q+5'r)/(4qr—4q—4r)
.0 ”.91.:[11.11.011.111 1.1:.I:.1::1133-311/111....::........:.:.I...I
...Q ..... 1 ............ l .01 .................................. ( .TT..”C.3)/.f1 ..........................
...Q. (1.3.7.2. ......... 1.0.] ................................. ( 9.1L. 7731.31.44 ........................
0,513.2... ........ [.01 ................................. ( .q.fc.7:..-..4)/.4 ........................
....1. ...-....2 ........ [Tl ............................ ( .t‘Tflrrji21/(47‘. fl: .8.) ...................
...1 ...1 ............. [Tl ............................ ( $313.71 .T.1)../.(.‘1?‘.ff..2.) ..................

1 q S _3 [r] (qzr + qr2 + (12 + r2 + 4qr + 3g + 3r)/(4qr + 4q + 41")

1 q22 [r] (q2r+qr2+q2+r2—2qr—3q—3r)/(4qr+4q+4r)

 

 

 

Table3.7. The case r>2 (|p] < 2 or |q| <2).

In Table 3.7, we do not need any computation to find bn(5): For any

row, we can use Remark 3.2.1. For example, in the 1“ row, we have an

overtwisted contact structure on the lens space L(m, -——1) for some m _>_ 1.

Therefore, the resulting contact manifold is not listed in Theorem 3.0.1, and

hence we must have bn(5) = 3.

In Table 3.8, we need to compute the binding number bn(é) for the rows

1, 3, 7, and 10. For the other rows, see Remark 3.2.1.

o If p = 1,q = —2,'r < -—2, {1:27. is an overtwisted contact structure on

L(|7‘ + 2],1). It is not hard to see that c1(§1,_2,,.) = [2] E erl—ZZ- Therefore,

82

 

p q 01(5) 6 WW) d3(€)
...1. ..... T .0 ............. l .21 .............................. (€23:Zr.i.1f1.>.[.(f1.r.f8) ....................
...1 ..... T .1.........9..€..{.9}. ...................................... 1 [3 .................................
...1.._....Q ............. [.01 ................................... ( .T.|.T.I.T..3)/.‘1 .............................

1 q S -—3 Hr” (qzr + qr2 + q2 + r2 +10qr + 9g + 97‘)/(4qr + 4g + 41‘)

1 q >1 [lrl] (q21‘+q1‘2 +q2 +73+4qr+3q+3r)/(4qr+4q+4r)
...0........0. ............. [.01 ..................................... ( .Tl’flifilfl ...........................
. . .0. ..... T. 1 ............. [.01 ..................................... ( .T. If I .T. .7.)[.‘1 ...........................
.0 £1.33? .......... [0] .................................... (9. 25?". .T. 8.1/4. ..........................
0....922 .......... [.01 .................................... (a. it 112.).[4 ..........................
.T1. .....3. ............ [[1].] ............................ (7.01.1371 .1“. .6).[ 1:40". 333.). .................
.T.1 ..... T1 ............ UT.” ............................ (.TTiTfiT. .T. 3)./(T171 .T. .2). .................
-1 q S - {lrll ((12? + qr2 - <12 — r2 + 6qr - 7q - 77‘)/(4:<1r — 4q - 4T)
—1 q 2 3 HT” ((127'+qr2 -(12-r2 -q-T)/(4qr-4q-4T)

 

 

 

Table 3.8. The case 7" < —2 (IpI < 2 or |q| < 2).

we immediately get bn(§1,q,_2) = 3 because C1(77|r+2|) = 0.

o If p = 1,q = 0,?“ < —2, the contact structure 51,0, on L(Irl, 1) and the
link 114,0”. are given in Figure 3.17. It is easy to see that c1(§1,0,r) = 0 E erl,
so we need d3(€1,0,,.): The corresponding linking matrix is on the left below.

Diagonalize £19,, to get the matrix on the right. Therefore,

0(£1,0,,—) = [7"] — I.

 

 

 

 

F—1—2—2 —2§—I) r1 0 0 0 :07
—2—1—2 . —2g—1 0 2 0 0 3 0
—2-—-2—1 ' E. 0 0 2. I E
£1,0,r — - —+ ' I ' 3
—1—2; - 2 0 5
i212.........T.2.T.1..3.T..1 0 0 .0 1/2‘ 0
L—l—l —1 OJ LOWO """""""" 0""¥f_3]

83

(b)

   

each
, framing -

 

 

 

 

 

 

Figure 3.17. (a) £10,, on L(|r|,1) for r < —2, (b) the link 114.0,?"

The system
£1,0,,.[b]T = [rot(Kl) T‘Ot(K2) - - - rot(Klql+1)]T = [1 - - - 1 0]T

has the solution [b] :2 [0-~-0 — 1], and so we obtain c2 = 0. Moreover,

x(X1,0,,.) = [r] + 2, and s = [7'] + 1. So we compute

d3(€1,0,r) = (—|r| + 3)/4 = d3(77|r|)

which implies that 51,0”. is isotopic to 77],] on L(|r|,1). Thus, bn(5)”) = 2
by Theorem 3.0.1.

o pr = 0,q = —1,7" < —2, the contact structure 50,-” on L(Irl, 1) and the
link L0,-” are given in Figure 3.18. Again we have c1(§0,_1,,.) = [O] E erl,
so we need to find d3(€0,_1,,): We diagonalize £071; and get the matrix on

the right below. So, we conclude that 0(£0,_1,,.) = |r] — 1.

84

 

 

 

 

 

F—1-2—2 —2§—1—1—P f1 0 0 0 §0 0 0]
—2—1—2 —25—1—1—1 0 2 0 0 30 0 0
—2—2—1 §. . O 0 2 :i.
coi—lir — —1_2[ . —1 .2 O
7.72:2.........T.2..T.1‘..T.1.T.1.T.1 Q...Q ........... 01/2000
——1—-1 ——1;0 0 0 0 0 . 0 z—30 0
—1-—1 —1§00—1 00 03020
[—1—1 —120 ~10J L0 0 0 so 0—1/2]
(1))
0 ._
—1 -—1
O 0 —

 

 

 

 

 

 

Figure 3.18. (a) £071}, on L(|rl, 1) for 7° < —2, (b) the link Log”.

The system
£0,_1,r[b]T = [rot(Kl) T0t(K2) - - - rot(Klq|+3)]T = [1 - . . 1 0 0 0]T

has the solution [b] = [0n 0 1 0 0], so we get 02 = 0. Also x(X1,q,_2) =
|r| + 4, and s = [r] + 3. So we compute d3(§0,_1,r) = (—|r] + 7)/4 79

(-l7‘| + 3)/4 = d3(77|r|) implying that éo,-i,r a: 77m on L(|T|,1)- Hence,

85

bn(éo,_1,r) = 3 by Theorem 3.0.1.
o If p = —1,q = 2,?" < —2, we have bn(§_1,2,,.) = 3 because cl(§_1,2,r) =
[lrl] E Z|r|+2' We compute cl(§_1,2,,.) as follows: We use the linking matrix

13-13,, to get the representation

H1(M) : < lab/'12: ' ' ' ,#|r|+3[ £—1,2,T[H][Tr[+3 : [ORA—[.3 >

= (#1|(|7"|+2)#1= 0) g Z|r|+2-

Moreover, using the relations given by 16-13,, we have m = 112- -- = ”M

(u,- ’s are the meridians as before). Therefore, we obtain

CHE—1,2,7) = I’D—1011 + - - ° + um) = PD’1(I7‘|M1) = “TH E Z|r|+2~

To finish the proof, in each table above we find each particular case for
(p, q, r) such that the corresponding contact structure épm. has binding
number 2. Note that the conditions on p, q, 7‘ given in the statement of the
theorem excludes exactly these cases. This completes the proof of Theorem

3.0.4. E]

3.3 Remarks on the remaining cases

Assume that r = 0, i1, |p| Z 2, lg] 2 2. We list all possible contact
structures in Table 3.9. These are the only remaining cases from which we
still get lens spaces or their connected sums. Notice that we have already

considered the cases (—2,q, 1), and (2,q, —1) in Tables 3.3 and 3.4, so we

86

do not list them here.

 

7" P q 01(5) 6 H2(M) (13(5)

.9..P...Z.?.. 0.2.? .......... l .0] ................................ (RT..‘1..T..‘1).[.‘1 ....................
.0,72..>...2...7.§:.2 ........ [.01 ................................ (21.91.2214. ....................
.0..P.§..-..07.§:?. ........ [.0] ................................ (Rf.0f.3l[f1 ....................

1 p22 (722 MD] (pzq+pqz+p2+q2-2pq-3p-3Q)/(4pq+4p+4q)
1 p 2 2 q < —2 l-pl (—p2q +77612 +792 + (12 + 4m + 312+ 3q)/(4pq + 479+ 4(1)
1 p < —2q < -2 [pl (p2q +70q2 +172 + (12 HOW + 9p + 9q)/(4pq + 477 + 4(1)
... ........... t ...... . .................................................................................

-1 17>? 61>? l-pl (p2q+pq2-p2-q2-6pq+5p+5Q)/(4pq-4p-4q)
~1p>2 (13—2 Hi] (p2q+79c12—p2—qz-p-q)/(4pq-4p-44)
“1P5 ‘2‘15 ‘2 [P] (p2q+pqz-p2-q2+6pq-7p-74)/(4pq-4p-4q)

 

 

 

 

Table 3.9. The case 7“ = 0,i1, |p| _>_ 2, |q| 2 2 (bn({) = 3 in each row).

As we remarked at the beginning of the chapter (after Theorem 3.0.4) that
one can obtain the complete list without any repetition: We first simply
find all distinct homeomorphism types of the manifolds which we found in
Table 3.2 through Table 3.9. Then on a fixed homeomorphism type we
compare the pairs (cl, d3) coming from the tables to distinguish the contact
structures.

Suppose now that M is a prime Seifert fibered manifold which is not a lens
space. Then as we remarked before we have [p] 2 2, lg] 2 2, and Ir] 2 2.
Then two such triples (p, q, 7‘), (p’,q’, 7") give the same Seifert manifold Y

if and only if

111_111
pqrfiq’ ”

87

and (p’.q’,7"’) is a permutation of (p, q,r) (see [JN], for instance). Notice
that. we can drop the first condition in our case. Switching p and q does not
change the contact manifold as we mentioned before. On the other hand, if
we switch 7" and p (or 7" and q), we might have different contact structures
on the same underlying topological manifold.

Another issue is that there are some cases where the first homology group
H1(Y(p, q, 7)) is not finite. Indeed, consider the linking matrix [I of the

surgery diagram given on the right in Figure 3.2 as below.

{0111\

17300
10q0

(1007")

The determinant det(£) = —7"(p+q) -pq = 0 implies that 7~ = ~pq/(p+q).

 

 

Thus, if 7' sé —pq/ (p + q), then H1(Y(p, q, 77)) is finite, and so d3(§p,q,,.) is
still computable since c1(§p,q,,.) is torsion. For instance, if p 2 2, q 2 2, 7' 2 2
or p g —2,q g —2,r g -—2, than det(£) 74 0, and so we can distinguish
the corresponding {mm by computing the pair (c1,d3). Whereas if the
sign of the one of p,q, 7" is different than the others’, then we might have
det(£) = 0. For instance, for the triples (4,4, ——2),(3,6, ——-2) and each
nonzero integer multiples of them, det(A) = 0. So more care is needed for
these cases.

We would like to end the chapter by a sample computation. Assume that

det(£) 75 0, and that r S 2, p 2 2, q g 2 (similar calculations apply for the

88

other cases). We compute the first homology of M 2 Y(p, q, 7“) as

H1(M) = ( #1412, ' " ,#p+q+|r|[ £p,q,r[/‘];+q+lr| = [OEHIHTI >

= < [11, Mr|+17 “PHTII R1’ R2’ R3 >

where the relations of the presentation are

R1: "(21T1—1)#1 — (p—1)H|rl+1 — ([QI+1)Hp+|r| : 0
R2: put-1+1 - ICIl/zpurl = 0

R3: —|7‘|u1 — PM|r|+1 = 0

While getting these relations, we also see that 111 = 112 ~ - - [1],] (recall ,ui’s
are the meridians to the surgery curves in the family corresponding to 7“ for
7' = 1, - . ~ , [7‘] ). Then using this presentation, and knowing that Cl (51,“) =

PD‘1(|7°|;11), we can evaluate (understand) c1(£p,q,,.) in H2(M) ”E H1(M).

Now if c1(§p,q,,.) E H 2(M ) is a torsion class, then we can also compute

d3(§p,q,,.) as follows: By solving the corresponding linear system we get

02 : FIGHT]
plql +plr| - [GMT]

 

Moreover, we compute 0(Xp,q,,.) = 0(£p,q,,.) = —p + [q] + |7~|,X(Xp,q,,.) =
p + [q] + [7"] + 1, and s = [q] + [7"] + 1. Hence, using Corollary 1.8.2, we
obtain

d (g )_ 879(17“+p2q+7927"+479612 +4617“2 —p7‘2 -q2r -pq —p7‘ -q7“
3 W _ 4pq + 4197" + 4q7‘ '

 

89

Chapter 4

Contact structures associated to

four-punctured sphere

In this chapter, we consider the contact structures for sg(§) = 0 and bn(é) S
4. We first focus on the mapping class group of the four-punctured sphere
in Section 4.1. In Section 4.2, we show certain families are holomorphically
fillable. We show the overtwistedness of certain families in Section 4.3 where
we also give alternative proofs of some results recently proved in [Y] (see
Lemma 4.3.3, Remark 4.3.5, Lemma 4.3.6).

Let S be any surface with nonempty boundary as before. We will stick
with the following convention: In Aut(S,BS), we will multiply a new el-
ement from the right of the existing (pre—introduced) word although we
compose the corresponding difeomorphisms of S from left. That is, if
a, 'y E Aut(S, BS), then (denoting the corresponding diffeomorphisms with

the same letters) we have
(0 - '00?) = (7 0 0X3?) = 7(0(w)) for a: E 5-
Let 2 be the four—punctured sphere obtained by deleting the interiors of

90

four disks from the 2—sphere S2 (see Figure 4.1). Let C1, C2, C3, C4 be the
boundary components of 23, and let a, b, c, d denote the simple closed curves
parallel to the boundary components C1, C2, C3, C4, respectively. Also con-

sider the simple closed curves 6, f, g, h in 2 given as in Figure 4.1.

 

 

 

Figure 4.1. Four—punctured sphere 2, and the simple closed curves.

Let qb E Aut(2,82) be any element. In Section 4.1, it will be clear that

we can write

¢ = arlbr2cr3d7‘4em1fnl . . . emsfns
for some integers m,- ’s and n,- ’s. Our main results are the following:

Theorem 4.0.1 If min{rk} _>_ max{—m,—n,0}, then (M¢,f¢) is
holomorphically fillable, where gb = a.“I)’"2c’"3d"4em1 f"1---ems f“ E

Aut(2,8§3) and m = Zlem, and n = 232171,.

91

Theorem 4.0.2 The contact structure 5,.) is overtwisted in the following

cases:
(1)77: < 0 for some 1:,

(2} rk = 0 for some k and min{m,n} < 0,

(3) min{rk} = 1, {r2 =1 or r4 =1}, min{m,n} < 0 and ran 2 2,
(4) min{rk} = 1, {r1 =1 0r r3 =1}, min{m,n} < 0 and ran 2 2,

where q5 = arlbr2c’3dr4emlfnl~-emsf"3 E Aut(2,82), m = Ef=1mi and

_ 8 _
n —— 2),-2171,.

4.1 Four—punctured sphere

For simplicity, we will denote the Dehn twist along any simple closed curve

by the same letter we use for that curve.

Definition 4.1.1 An element d) E Aut(2,82) is said to be in reduced

form if there exists an unique integer 0 S s such that gb can be written as

Cb : arlbr2cr3dr4em1fnlem2fn2 . . . emsfns

where rk,m,-,n,- are all integer for 1 S k S 4, 1 g i _<_ s with possibly ml

or n, zero.

Lemma 4.1.2 Any element qt 6 Aut(2,62) can be written in reduced

form.

92

Proof: From braid group representation of full mapping class group,
we know that the mapping class group Aut(2,82) can be generated by
Dehn twists along the simple closed curves a, b, c, d, e, f, g, h given in Figure
4.1 (see [Bi] for details). Therefore, any element qt of Aut(2,8§3) can be
written as a word consisting of only a, b, c, d,e, f, g,h and their inverses.
Since a, b, c,d, are in the center of Aut(S, 62), we can bring them to any
position we want. For the second part including 6 and f, we use the well-
known Lantern relation (also known as 4-holed sphere relation). In terms
of our generators we will use two different Lantern relations. Namely, we

have
ge f = abcd and h f e = abcd.

These give 9 = abcdf'le"l and h = abcde‘lf’l. Therefore, we can
exchange any power of g and h in the word defining (t by some products
of a, b, c, d, e‘1,f‘1 (and a‘1,b'1,c“1,d‘1,e,f for negative powers of g
and h). Combining (and canceling if there is any) the powers of e and f,
and commuting the generators a,b, c,d, we get the reduced form of gb as

claimed. [:1

From now on, we will always consider the elements of Aut(E, BE) in their
reduced forms. In the following, we will say that two monodromy elements
h1,h2 E Aut(S,8S) on the same surface S are equivalent if the contact
manifolds (M(5,h1),§(5,h1)), (M(5,h2),§(5,h2)) are contactomorphic. We will

denote this equivalence by hl ~ hg.

93

Theorem 4.1.3 Let 45 = arlbr2c""”d'"‘1emlf’”emf"2 - - - emsfns E Aut(S, (92) be
as before. Consider the element (15’ = arlbr2cr3df4emf" where m = Zf=1mi

and n = Ef=1n,. Then qt) ~ (15’, i.e., there is a contactomorphism

(M(2,¢),€(2,¢)) g (M(2,¢I),§(2,¢'))-

Proof: First consider only the last parts (150 = em1 fnlemz f"2 - - - ems f’”
and (b6 = emf" of gb and (75’. We will show that (M(2’¢0),€(§3,¢0)) and
(M(2,¢,70), €(E,¢IO)) are contactomorphic (indeed they are isomorphic as open
books). We will induct on 3. All the equivalences in the induction follow

from Theorem 1.1.1 and Theorem 1.5.1.

For 3 = 2, we have

N eml

r——~—\
emlfnlem2fn2 ~ f—nl . emlfnl em2fn2 . fnl ~ eml+m2fnl+n2

proving the first step of the induction. Now assume that the result is true
for s — 1, ie., emlfmemf"2 - - - em1s‘llf”(3‘1) ~ efiffi where ‘rfi = Zillm,
and n = Zillni. Then

em] fnl . _ . em(s—l)fn(s—1) . emsfns N eT—n‘ffi . emsfns
m

N6

r—A—3
Nf-Tl' . efifn emens . f‘fi ~ efi+m3ffi+ns

which proves the result for 3. Therefore, we have showed that 050 ~ ([76.
Now adding the same word a’"1b’"2c’"3d’4 to both 050 and (bf, give contacto-

morphic manifolds by Theorem 1.5.1, so (75 ~ (15’ (Recall that a1'1b'°2c’"3d"4 is

94

a central element of Aut(S, 82)). E]

4.2 Holomorphically fillable contact structures

In this short section, we will prove Theorem 4.0.1 using the lantern relation.
Proof: [Proof of Theorem 14.0.1] By Theorem 4.1.3, we will prove the
statement for 5,3: where (75’ = a"1b"‘icr3dT 4e’" f". We assume that m S n,
the other case is similar. If m 2 0 then the result follows immediately
from Theorem 1.4.2. Otherwise we first find a monodromy d for which
(M(2,¢I),€(E,¢I)) is contactomorphic to (M(2,3)15(2,3)) using Theorem 4.1.3
and then use the Lantern relations to write the monodromy q; as a product

of positive Dehn twists as follows:

—m times

 

 

(5 : arlbr2cr3d’4(e_1f_1)(e—1f’l)~ . . (e—lf—T) fn—m
—m times
= arl+mbr2+mcr3+mdr4+m (abcde'lf_l) . . -(abcde’1f’l) fn—m
—m times

: ar1+mbr2+mcr3+mdr4+m ’h . h . . . h‘ fn—m_

Theorem 1.4.2 implies that 5,]; is holomorphically fillable. Hence, 64,7 and

6,, are also holomorphically fillable. El

95

4.3 Overtwisted contact structures

Among the contact structures Q, with (15 E Aut(E, 02), we want to distin-

guish overtwisted ones. First, we prove three lemmas.

Lemma 4.3.1 Let S be a planar hyperbolic surface with geodesic boundary
8S = Ufilei, l 2 4. Suppose h E Aut(S,8S) and there is a properly
embedded are 7 starting at :1: E C,, ending at Cj such that M7) is to the

left of '7 at .7: and i 55 j. Then (h - D5)('y) is to the left of '7 at :c E C,- for

any curve 6 parallel to Ck with 1: 7E i.

Proof: Isotoping if necessary, we may assume that ’y and My) intersect
minimally. We need to analyze two cases:

Case 1. Suppose k 75 j. Then we may assume 706 = (l, and so h(7)fl6 =
0. That is , D5 fixes both 7 and h(7). This implies that D5(h(’y)) = h(7)

is to the left of 'y.

 

 

 

 

\\
“Q
3“
A
:3,
a

\\

 

 

 

Figure 4.2. h(7) is to the left of '7 (Left and right sides are identified).

96

Case 2. Suppose k = j. First note that h 74 ids since h is not a right

veering. Therefore, there exists a region R C S such that

1. R is an embedded disk punctured r—times for some 0 < r < m — 2,

and
2. 8R C 7Uh(7) UBS.

Let Ci1,- .. ,Cir be the common components of 3S and ER. We may as
sume that 8R contains the common initial point a: and the first intersection

point y (of 7 and h(7)) coming right after it (See Figure 4.2).

 

 

 

 

\\
\\
(Q
\
...b
A
@T
A
~2
v
V
\x
\\

 

 

 

 

 

Figure 4.3. D5(h(7)) is to the left of 7 (Left and right sides are identified).

Since the Dehn twist D5 is isotopic to the identity outside of a small
neighborhood of 6, the image R’ = 05(R) is isotopic to R. In particular,
BR’ 0 D5(h(7)) is to the left of 8R’ 0 7 (see Figure 4.3). Note that
D5(h(7)) and 7 are also intersecting minimally. Therefore, we conclude
that (h - D5)(7) = D5(h(7)) is to the left of 7. El

97

The following corollary of Lemma 4.3.1 is immediate with the help of The-

orem 1.7.2.

Corollary 4.3.2 Let S be a planar hyperbolic surface with geodesic bound-
ary 8S = Ufilei, l 2 4. Suppose h E Aut(S,BS) is not right veering with
respect to C,- for some i, and so the contact structure {(S,h) is overtwisted.
Then the contact structure compatible with (S,h - D5,“) is also overtwisted
for any k E Z+ and for any curve (5 parallel to the boundary component

which is different than C,. E)

Let us now interpret the notion of right-veering in terms of the circle at infin-
ity as in [HKMl]. Let S be any hyperbolic surface with geodesic boundary
BS. The universal cover 7r : S —> S can be viewed as a subset of the
Poincaré disk D2 = H2 U SCI”. Let C be a component of 8S and L be a
component of 7r“1(C). If h E Aut(S,6S), let h be the lift of h that is the
identity on L. The closure of S in D2 is a starlike disk. L is contained in
(9S . Denote its complement in 8S by Loo. Orient L00 using the boundary
orientation of S and then linearly order the interval L00 via an orientation-
preserving homeomorphism with IR. The lift h induces a homeomorphism
hoO : LOO ——> Loo. Also, given two elements a, b in H omeo+ (IR), the group of
orientation-preserving homeomorphisms of R, we write a 2 b if a(z) Z b(z)
for all z E R and a > b if a(z) > b(z) for all z E IR. In this setting, an

element h is rigth-veering with respect to C if id _>_ hoo. Equivalently, if a

98

is any properly embedded curve starting at a point 07(0) E C, and 62 is the

lift of a starting at the lift (1(0) E L of :12, then we have
h(oz) is to the right of a 42> 67(1) 2 hoo(d(1))

Therefore, h is not right-veering with respect to C if there is an arc a

starting at C such that we have d(1) < hoo(6z(1)).

Lemma 4.3.3 Let S be any hyperbolic surface with geodesic boundary 6S .
Suppose h E Aut(S, BS) and there is a properly embedded are 7 starting at
:r E C C (9S such that h(7) is to the left of 7 at :5. Then (h ~ D;I)(7) is

to the left of 7 at a: E C for any simple closed curve a in S.

Proof: Write a for D0. Fix the identification of LC>0 with IR as above.

Consider the lift 7' and induced homeomorphisms hoe, 000, 0;} : L00 —> L00.

1

Since 0‘ -a = ids, we have

—1 -—1

(a «7).,O = 000 o 000 = (id5)oo.

—1

00 must map any point in L00 to its left because a is right-

Therefore, a
veering. In particular, (h - 0‘1)00(7(1)) = ogol(hoo(7(1))) is to the left
of hoo(7(1)) which is (by the assumption) to the left of 7(1). That is,
(h - 0‘1)00(7(1)) > hoo(7(1)) > 7(1). Hence, (h - 0‘1)(7) is to the left of 7.

I]

Corollary 4.3.4 Let S be a planar hyperbolic surface with geodesic bound-
ary 8S = U[:1C,-, l 2 4. Suppose h E Aut(S, BS) is not right veering with

99

respect to C,- for some i, and so the contact structure {(S,h) is overtwisted.
Then the contact structure compatible with (S, h - D01“) is also overtwisted

for any 17 E Z_ and for any simple closed curve a in E. E]

Remark 4.3.5 The idea used in the proof of Lemma 4.3.3 gives a simple
proof for Lemma 6. of [Y]. Moreover, the following lemma is given as
Lemma 5 in [Y]. We want to give a difierent proof for it using the idea of

the circle at infinity.

Lemma 4.3.6 Let S be a hyperbolic surface with geodesic boundary, and
let h E Aut(S,3S) be a right-veering difleomorphism. Then h’ = aha“1 is

right-veering for any a E Aut(S, 03).

Proof: Clearly, it is enough to consider the case when 0 is a single Dehn
twist. First, assume that 0‘ is a positive Dehn twist. We need to show that
h’ is right-veering with respect to any boundary component of S. We will
use the notations introduced in the previous paragraph. So fix the boundary
component C, and an identification of L00 with R as above. Let a be any
properly embedded curve in S starting at a point 07(0) E C. Consider the
lift 51 and induced homeomorphisms hgo,hoo,ooo,o;ol : Loo —> Loo. From

their definitions we have

N ~

~

h' (54(1)) = 5151(1)) = aha-101(1)) = 5h0‘1(d(1)) = UoohooUJOl(5t(1))

00

100

Suppose that ogol(d(1)) = a E LC,O and hoo(a) = b E Loo. Then since

00.7)) = ((a-1>‘1>..<b) = (aw—Rb),

b must be mapped (by 000) to a point in LOO which is to the right of 67(1)

as we illustrated in Figure 4.4. Equivalently, 57(1) 2 awhooogo1 (61(1)) =

A
I I
o

the region the region
31 to the left 5, to the right

aV/Nb 62(1) goohooasou» L...

..............................................................................

Figure 4.4. The point (13(1) E L00 z IR, and how it is mapped to the right of
itself.

hfx,(57(1)) implying that h’ is right-veering with respect to C. The proof of
the case when a is negative Dehn twist uses exactly the same argument, so
we omit it. C]

Now we can characterize the overtwisted structures stated in the introduc-

tion.

4.4 The proof of Theorem 4.0.2

Proof: By Theorem 4.1.3, we will prove the statements for 5,7,7 where
as! : arlbr2cr3dr4emfn.

101

To prove (I), consider the properly embedded curves 071,072, £13,074 starting
at the boundary components C1, C2, C3, C4, respectively, and their images

under gb’ as given in Figure 4.5. In all the pictures, we are assuming m > 0,

 

 

(Wall (#012)

. 02 co
r = —1 O ' O ' r =—1
1 .038 O@ 2
¢I(a3) a4
7‘3=—1 O (0: iii] C— O " O 7‘4=—1
'— .. ¢/(a4)

 

 

 

 

 

 

 

 

 

 

21:;

 

 

 

 

 

 

 

Figure 4.5. The curves 0;, and their images under gb’ in E.

n > 0, and rk = —1 (otherwise the fact that 3’ is not right-veering with
respect to Ck is even more obvious). We can see from the pictures that
if rk < 0 for some k, then (flak) is to the left of (1;, , so 43’ is not right-
veering which implies by Theorem 1.7.2 that {97,7 is overtwisted. Note that
in any picture in Figure 4.5, we are taking all the other rk’s to be zero.
However, even if (15’ has a factor of some positive power of Dehn twist along
the boundary component other than Ck, (flak) is still left to the m, at their
common starting point by Lemma 4.3.1, so {p is overtwisted by Corollary
4.3.2.

To prove (2), consider the properly embedded curves ,61, ,82, [33, 64 starting

at the boundary components C1, C2, C3, C4, respectively, and their images

102

 

 

 

c To

¢’(fi2)

 

 

 

 

 

 

 

   
 

 

 

¢'(53)'=¢'(54)

 

 

 

 

 

 

 

 

 

Figure 4.6. The curves 6;, and their images under (75’ in 2.

under gb’ as given in Figure 4.6. In all the pictures, we are assuming m =
—1, n > 0, (again otherwise the fact that d’ is not right-veering with
respect to Ck is even more obvious). We can see from the pictures that if
rk = 0 for some k, then 45(6),.) is to the left of 6;, , so (75’ is not right-veering
which implies again by Theorem 1.7.2 that 75¢, is overtwisted. Again, in all
the pictures, we consider all the other rk’s to be zero, and if gb’ has a factor
of some positive power of Dehn twist along the boundary component other
than Ck, (b’wk) is still left to the 6;, at their common starting point by
Lemma 4.3.1. Therefore, (,9, is overtwisted by Corollary 4.3.2.

To prove (3), consider the curve 7 running from C2 to C; as in Figure
4.7. In the left picture each rk = 1, m = —2, n = —1, and in the right one
each rk = 1,m = —1,n = —2. Clearly, the image (77(7) is to left of 7 at

both their common endpoints on C2 and C4, so Sp (ct’ = abcde’2f‘l or

103

abcde‘1 f ‘2) is overtwisted. In both cases, if we take r1, r3 and only one of
r2 and r4 to be any positive integer, 75¢: is still overtwisted by Lemma 4.3.1
and Corollary 4.3.2. Moreover, for m g —3,n S —3 in both cases, 6,57 is

overtwisted by Lemma 4.3.3 and Corollary 4.3.4.

 

 

¢’(7)
O O O 0
6(7) 7 7

 

 

 

 

 

 

Figure 4.7. The curve 7 and its images under two possible (15’ in E.

The proof of (4) is similar to that of (3), so we will omit it. D

104

BIBLIOGRAPHY

105

BIBLIOGRAPHY

[AO] S. Akbulut, B. Ozbagci, Lefschetz fibrations on compact Stein surfaces,
Geom. Topol. 5 (2001), 319-334 (electronic).

[Bi] J. S. Birman, Braids, links and mapping class groups, Annals of Math-
ematics Studies, No. 82. Princeton University Press, Princeton, NJ;
University of Tokyo Press, Tokyo, 1974.

[DC] F. Ding and H. Geiges, A Legendrian surgery presentation of contact
3-manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3,
583—598.

[DGS] F. Ding, H. Geiges and A. I. Stipsicz, Surgery diagrams for contact
3—manif0lds, Turkish J. Math, 28 (2004) no. 1, 41-74.

[EG] Y. Eliashberg and M. Gromov Convex symplectic manifolds, Proc. of
Symp. in Pure Math. 57 (1991), 135—162.

[Etl] J. B. Etnyre, Planar open book decompositions and contact structures,
Int. Math. Res. Not. 2004, no.79, 4255—4267.

[Et2] J. B. Etnyre, Lectures on open book decompositions and contact struc-
tures, Floer homology, gauge theory, and low-dimensional topology,
103-141, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006.

[Et3] J. Etnyre, Introductory Lectures on Contact Geometry, Topology and
geometry of manifolds (Athens, GA, 2001), 81—107, Proc. Sympos. Pure
Math., 71, Amer. Math. Soc., Providence, RI, 2003.

[E0] J. Etnyre, and B. Ozbagci, Invariants of Contact Structures from Open
Books, arXiv:math.GT/ 0605441, preprint 2006.

106

[Ga] D. Cabai, Detecting fibred links in S3, Comment. Math. Helv.,
61(4):519-555, 1986.

[Ce] H. Geiges, Contact geometry, Handbook of differential geometry. Vol.
II, 315—382, Elsevier/North—Holland, Amsterdam, 2006.

[Cd] N Goodman, Contact Structures and Open Books, PhD thesis, Uni-
versity of Texas at Austin (2003)

[Ci] E. Giroux, Ce’ome’trie de contact: de la dimension trois vers les di-
mensions supe’rieures, Proceedings of the ICM, Beijing 2002, vol. 2,
405-414.

[Gm] R. E. Gompf, Handlebody construction of Stein surfaces, Ann. of
Math. 148 (1998), 619—693.

[GS] R. E. Gompf, A. I. Stipsicz, 4-manifolds and Kirby calculus, Graduate
Studies in Math. 20, Amer. Math. Soc., Providence, RI, 1999.

[HKMl] K. Honda, W. Kazez and G. Matic, Right-veering difieomorphisms
of compact surfaces with boundary, Invent. Math. 169 (2007), no. 2,
427—449.

[HKM2] K. Honda, W. Kazez and C. Matic, 0n the contact class in Hee—
gaard Floer homology, arXivzmath/ 0609734, preprint 2006.

[Ho] K. Honda, 0n the classification of tight contact structures -1, Geom.
Topol. 4 (2000), 309—368 (electronic).

[J N] M. Jankins and W. D. Neumann, Lectures on Seifert manifolds, Bran-
deis Lecture Notes 2, Brandeis University (1981).

[MS] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Ox-
ford University Press, 1995.

[LP] A. Loi, R. Piergallini, Compact Stein surfaces with boundary as
branched covers of B4, Invent. Math. 143 (2001), 325—348.

[Ly] H. Lyon, Torus knots in the complements of links and surfaces, Michi-
gan Math. J. 27 (1980), 39—46.

107

[Or] P. Orlik, Seifert Manifolds, Lecture Notes in Mathematics (1972),
Springer-Verlag, Berlin - Heidelberg- New York.

[OSt] B. Ozbagci, A. I. Stipsicz, Surgery on contact 3-manifolds and Stein
surfaces, Bolyai Society Mathematical Studies, 13 (2004), Springer-
Verlag, Berlin.

[OSz] P. Ozsvath, Z. Szabo, Heegaard Floer homology and contact struc-
tures, Duke Math. J. 129 (2005), no. 1, 39—61

[Pl] O. Plamenevskaya, Contact structures with distinct Heegaard F loer in-
variants, Math. Res. Lett., 11 (2004), 547-561.

[St] A. I. Stipsicz, Surgery diagrams and open book decomposition of contact
3-manifolds, Acta Math. Hungar, 108 (1-2) (2005), 71-86.

[SW] S. Sarkar and J. Wang, A combinatorial description of some Heegaard
F loer homologies, preprint (2006), math/ 0607777.

[TW] W. P. Thurston, H. E. Winkelnkemper, On the existence of contact
forms, Proc. Amer. Math. Soc. 52 (1975), 345—347.

[Y] E. Yilmaz, A Note on Overtwisted Contact Structures, arXiv:0712.4379,
preprint 2008.

108

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