CONCAVE FILLINGS AND BRANCHED COVERS

By

Kaveh Kasebian

A DISSERTATION

Submitted to

Michigan State University

in partial ful(cid:12)llment of the requirements

for the degree of

Mathematics - Doctor of Philosophy

2018

ABSTRACT

CONCAVE FILLINGS AND BRANCHED COVERS

By

Kaveh Kasebian

This dissertation contains two results. The (cid:12)rst result involves concave symplectic struc-

tures on a neighborhood of certain plumbing of symplectic surfaces, introduced by D. Gay.

We draw the contact surgery diagram of the induced contact structure on boundary of a

concave (cid:12)lling, when the induced open book is planar. We show that every Brieskorn sphere

admits a concave (cid:12)lling in the sense of D. Gay and the induced contact structure on it is
overtwisted. We also show that in certain cases a ((cid:0)1)-sphere in Gay’s plumbing can be

blown down to obtain a concave plumbing of the same type. The next result examines the

contact structure induced on the boundary of the cork W1, induced by the double branched

cover over a ribbon knot. We show this contact structure is overtwisted in a speci(cid:12)c case.

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Contact structures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Open book decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Criteria for overtwistedness
2.4 Contact structure induced by the branched cover
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
2.5 Homotopy invariants of contact structures
2.6 Concave (cid:12)llings of contact manifolds
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6.1 Concave (cid:12)lling of positive plumbings

2.7 Corks

Chapter 3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Concave (cid:12)llings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A contact structure on the boundary of the cork . . . . . . . . . . . . . . .

3
3
7
8
11
12
15
16
19

20
20
28

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

iii

LIST OF FIGURES

Figure 2.1:

front projection of two Legendrian knots . . . . . . . . . . . . . . . . . .

Figure 2.2: positive stabilization of an open book . . . . . . . . . . . . . . . . . . . .

5

7

Figure 2.3: A sobering arc

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

10

Figure 2.4: Seifert surface of a (4; 4)-torus link . . . . . . . . . . . . . . . . . . . . .

12

Figure 2.5: construction of concave (cid:12)lling . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.6: The family of corks Wn . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Figure 3.1: a concave plumbing graph and its corresponding open book . . . . . . .

21

Figure 3.2: A special positive plumbing graph . . . . . . . . . . . . . . . . . . . . . .

22

Figure 3.3:

the lantern relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 3.4:

removing a positive Hopf band . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 3.5: Blowing down and the corresponding open books

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

23

Figure 3.6: plumbing diagram of a Brieskorn manifold . . . . . . . . . . . . . . . . .

24

Figure 3.7: A positive plumbing graph for (cid:6)(2; 3; 5)

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

24

Figure 3.8:

the open book on (cid:6)(2; 3; 5)

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

24

Figure 3.9: A Legendrian diagram for (cid:6)(2; 3; 5) . . . . . . . . . . . . . . . . . . . . .

25

Figure 3.10: The (cid:0)E8 plumbing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Figure 3.11: A branch in the plumbing . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Figure 3.12: modifying the open book . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Figure 3.13: Cork as branched cover over a ribbon disk . . . . . . . . . . . . . . . . .

29

Figure 3.14: (cid:11) and  ((cid:11)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

iv

Chapter 1

Introduction

This Dissertation contains two results. The (cid:12)rst result involves concave (cid:12)llings of contact

3-manifolds. In [8] D. Gay introduced a procedure for handle-by-handle construction of a

concave symplectic structure on a neighborhood of a certain type of plumbing of symplectic

surfaces- called a positive plumbing- in a symplectic manifold (X; !). His method also

speci(cid:12)es the induced contact structure on the boundary of this type of plumbing by its

compatible open book. We demonstrate how to draw the contact surgery diagram of these

contact structures in the case when the compatible open book is planar (by looking at the

induced open book on the Brieskorn sphere (cid:6)(2; 3; 5) as an example). A natural question is

which 3-manifolds can be presented in this way as boundary of concave (cid:12)llings. We show

that every Brieskorn manifold has a surgery diagram as a positive plumbing and therefore

its neighborhood in a symplectic manifold carries a concave structure. We show that the

contact structure induced on Brieskorn spheres by their concave (cid:12)llings is always overtwisted.
We also show that in certain cases a ((cid:0)1)-framed sphere in the concave (cid:12)lling of a contact

3-manifold can be blown down to obtain a smaller concave (cid:12)lling of the same contact 3-

manifold.

Any 3-manifold presented as a regular p-fold branched cover of (S3; (cid:24)st) over a transverse

knot K, can be assigned a natural contact structure induced by the cover. Our next result

involves such a contact structure on the boundary of the Akbulut cork W1, induced by

1

considering the boundary as a double branched cover over a ribbon knot K and considering

a speci(cid:12)c transverse realizations of K obtained by braiding. We show that for certain braiding

of the ribbon knot K the induced contact structure is overtwisted. Our motivation for this

problem was to consider a naturally induced contact structure on @W1 and examine its

tightness. Using the right-veering criteria for monodromy of open books, we show that in a

certain case the induced contact structure is in fact overtwisted.

2

Chapter 2

Background

In this chapter we review the background results that are needed for our main results.

2.1 Contact structures

For an introduction to contact structures and open books on 3-manifolds, the reader is

advised to [1].

De(cid:12)nition 2.1.1. Suppose Y is a (2n + 1)-dimensional manifold. A 1-form (cid:11) 2 Ω1(Y )
is called a contact form if (cid:11) ^ (d(cid:11))n is nowhere zero. A 2n-dimensional distribution (cid:24) is

called a contact structure if it locally can be written as (cid:24) =ker(cid:11).

We will only work with contact 3-manifolds and from now on most of our de(cid:12)nitions and

examples involving contact manifolds will be limited to this case only.

Example 2.1.2. The standard contact structure (cid:24)st on R2n+1 with coordinates

(x1; y1; :::; xn; yn; z) is given as ker(dz + (cid:6)n

1 xidyi).

Example 2.1.3. The standard contact structure (cid:24)

′
st on S3 thought of as the unit sphere
in C2 is de(cid:12)ned as (cid:24)st = T S3 \ i(T S3). Using coordinates (r1ei(cid:18)1; r2ei(cid:18)2) on C2, we can also
describe this contact structure as ker(r2

1d(cid:18)1 + r2

2d(cid:18)2).

De(cid:12)nition 2.1.4. Two contact 3(cid:0)manifolds (Y; (cid:24)) and (Y

phic if there is a diffeomorphism f : Y ! Y

′

such that f⋆((cid:24)) = (cid:24)

′

′

; (cid:24)
′

) are called contactomor-
′

′

. If (cid:24) =ker(cid:11) and (cid:24)

=ker(cid:11)

,

this is equivalent to existence of a nowhere zero function g on Y such that f

(cid:3)

′

((cid:11)

) = g(cid:11).

3

Two contact structures (cid:24) and (cid:24)
′
tomorphism h : (Y; (cid:24)) ! (Y; (cid:24)

) which is isotopic to the identity.

′

on a manifold Y are said to be isotopic if there is a contac-

Example 2.1.5. One can show that for a point p 2 S3, (S3(cid:0)fpg; (cid:24)

′
st) is contactomorphic

to (R3; (cid:24)st).

Example 2.1.6. Let (cid:11)

. We call (cid:24)sym the
symmetric contact structure on R3 (One can see that the contact planes are symmetric with

= dz +xdy(cid:0)ydx = dz +r2d(cid:18) and (cid:24)sym =ker(cid:11)

′

′

respect to the z-axis). This contact structure is contactomorphic to the standard contact
structure (cid:24)st on R3. Refer to [1] for more details. We will come back to this contact structure

later in 2:1:9:.

De(cid:12)nition 2.1.7. Suppose that (Y; (cid:24)) is a given contact 3-manifold. A knot K (cid:26) Y is
Legendrian if the tangent vectors T K satisfy T K (cid:26) (cid:24). In other words (cid:11)(T K) = 0 for the
contact 1(cid:0)form (cid:11) de(cid:12)ning (cid:24). The knot K is transverse if T K is transverse to (cid:24) along the

knot K, i.e. if (cid:11)(T K) is nonzero. The contact framing of a Legendrian knot is de(cid:12)ned by

he normal of (cid:24) along K. Equivalently, we can take the framing obtained by pushing K off

in the direction of the vector (cid:12)eld transverse to K which stays inside the contact planes.

This framing is called the Thurston-Bennequin framing of the Legendrian knot K denoted

by tb(K). Another invariant of a Legendrian knot, rotation number rot(K) can be de(cid:12)ned

by trivializing (cid:24)st along K and then taking winding number of T K. For this invariant to

be well-de(cid:12)ned we need to orient K and then the result will change sign when orientation is

reversed.

We can study Legendrian knots in standard contact R3 (or S3) via their front projection.
Namely, for (cid:24)st =ker(dz + xdy) and a Legendrian knot K (cid:26) (R3; (cid:24)st), we consider its

projection onto the yz-plane. Notice that the front projection has no vertical tangencies as
= (cid:0)x ̸= 1. For the same reason, at a crossing the strand with smaller slope is in front.

dz
dy

4

For instance the following is the front projection of two different Legendrian unknots:

Figure 2.1: front projection of two Legendrian knots

Lemma 2.1.8. For a Legendrian knot K 2 (R3; (cid:24)st), we have the following formula for

the Thurston-Bennequin number of K: tb(K) = w(K) (cid:0) 1
2

c(K), where w(K) is the writhe

of K and c(K) is the number of cusps in the front projection of K.

Proof. We note that the vector

the linking number lk(K; K

′

) where K

@
@z

is transverse to (cid:24) = ker(dz +xdy) so that tb(K) is just
′

is the push-off of K in the direction of this vector.

Now for the linking number we count the number of crossings of K
′

easy to see that a self-crossing of K will result in a crossing of K

′

and K with sign. It is

and K of the same sign.

A cusp on the left will give a negative crossing of K

′

under K and a cusp on the right will

give a crossing of K

′

are equal.

over K. The result follows because the number of left and right cusps

Lemma 2.1.9. The rotation number rot(K) of a Legendrian knot K is given by the

formula: rot(K) =

1
2

(cd(K) (cid:0) cu(K)), where cd and cu are the number of down and up

cusps in the projection.

Proof. The vector (cid:12)eld

@
@x

gives rise to a trivialization of (cid:24)st, hence the rotation number

can be counted as the winding number with respect to this vector (cid:12)eld. We have to count

the number of times the tangent of K passes the vector (cid:12)eld as we traverse K. De(cid:12)ne l(cid:6)

(resp. r(cid:6)) as the number of left (resp. right) cusps where the knot K is oriented upward or
downward. Then we can see that rot(K) = l(cid:0) (cid:0) r+. Counting with respect to (cid:0) @
rot(K) = r(cid:0) (cid:0) l+ and taking the average gives the result.

we get

@x

5

tb(K) = 0 (cid:0) 1
2

For instance, for the two Legendrian knots on the left and the right in (cid:12)gure 2:1 we have
) = (cid:0)1 (cid:0) 1
(2) = (cid:0)2 respectively. Similarly for the rotation
2
′
) = (cid:6)1 depending on the orientation on the knot

numbers we have: rot(K) = 0 and rot(K

(2) = 1 and tb(K

′

′

.

K

De(cid:12)nition 2.1.10. An embedded disk D (cid:26) (Y; (cid:24)) is an overtwisted disk if @D = K

is a Legendrian knot with tbD(K) = 0, i.e. the contact framing of K coincides with the

framing given by the disk D. A contact manifold (M; (cid:24)) is called overtwisted if it contains

an overtwisted disk; (Y; (cid:24)) is called tight otherwise.

According to a fundamental result of Eliashberg, overtwisted contact structures on 3-

manifolds can be classi(cid:12)ed up to homotopy of plane (cid:12)elds, as in the following theorem:

Theorem 2.1.11. Two overtwisted contact structures are isotopic, if and only if they

are homotopic as oriented 2-plane (cid:12)elds. Moreover, every homotopy class of 2-plane (cid:12)elds

contains an overtwisted contact structure.

Therefore, the classi(cid:12)cation of overtwisted contact structures reduces to a homotopy

theoretic problem which is not hard to solve. For more discussion on the above theorem the

reader can consult [2].

We will need to represent transverse knots as braids. Let us consider the symmetric

version of the standard contact structure (S3; (cid:24)sym) with (cid:24)sym = ker(dz + xdy (cid:0) ydx).

Given a closed braid B braided about the z-axis, we can isotopy it through closed braids
g, away from the

so that it is far from the z-axis. As (cid:24)sym = spanfx

+ y

; x

@
@z

(cid:0) @
@y

@
@x

@
@y

z-axis the planes that make up (cid:24)sym are almost vertical. Thus the closed braid B represents

a transverse knot. The opposite of the above theorem is also true:

Theorem 2.1.12 Any transverse knot is transversely isotopic to a closed braid.

Refer to [9] for a proof of the above theorem.

6

2.2 Open book decompositions

De(cid:12)nition 2.2.1. Suppose there is a link L in a 3-manifold Y that the complement Y (cid:0) L
(cid:12)bers as (cid:25) : Y (cid:0) L ! S1 such that (cid:12)bers are interiors of Seifert surfaces for L. Then (L; (cid:25))
(cid:0)1(t) is called a page and L

is called an open book decomposition of Y . Each (cid:12)ber (cid:6) = (cid:25)

the binding of the open book.The monodromy of (cid:12)bration (cid:25) is called the monodromy of the

open book decomposition.

A theorem of Alexander states that every 3-manifold admits an open book decomposition.

Refer to [1] or [2] for a proof and further discussion.

Example 2.2.2. Consider S3 as the unit circle in C2. De(cid:12)ne D = f(r1; (cid:18)1; r2; (cid:18)2) 2 S3 :
r2 = 0g. The (cid:12)bration (cid:25) : S3 (cid:0) D ! S1 given by (cid:25)((r1; (cid:18)1; r2; (cid:18)2)) = (cid:18)2 gives rise to an
open book on S3 with page a disk and monodromy equal to the identity.

De(cid:12)nition 2.2.3. Given an open book decomposition ((cid:6); ϕ), we attach a 1-handle to

′
the surface (cid:6) connecting two points on @(cid:6) to obtain a new surface (cid:6)

. Let (cid:11) be a closed

′

curve in (cid:6)

going over the new 1-handle once, as in the following (cid:12)gure. The new open book

′
((cid:6)

; ϕot(cid:11)) is called a positive stabilization of the original open book (where t(cid:11) denotes a

positive Dehn twist about (cid:11)).

Figure 2.2: positive stabilization of an open book

De(cid:12)nition 2.2.4. An open book decomposition is said to be compatible with the contact

structure (cid:24) on Y if (cid:24) can be represented by a contact form (cid:11) such that the binding is a

transverse link, d(cid:11) is a volume form on every page and orientation of the transverse binding

7

induced by (cid:11) agrees with boundary orientation of the pages.

The conditions (cid:11) > 0 on the binding and d(cid:11) > 0 on the pages can be thought of
strengthening of the contact condition (cid:11) ^ d(cid:11) > 0 in the presence of an open book on M .

Example 2.2.5. We can see that the trivial open book for S3 in the previous example

is compatible with (cid:24)st as follows: the tangent to the binding is given by

@
@(cid:18)1

and the contact

form is d(cid:18)1 restricted to r2 = 0. Therefore the binding is transverse to the contact structure
1d(cid:18)1) = 2r1dr1 ^ d(cid:18)1 is a

(cid:24)st. The contact form restricted to a page is r2

1d(cid:18)1 and thus d(r2

volume form.

The following theorem of Giroux states that open books up to positive stabilization

correspond to contact structures up to isotopy:

Theorem 2.2.6. (a) For a given open book decomposition of Y there is a compatible

contact structure (cid:24) on Y . Contact structures compatible with a (cid:12)xed open book decomposition

are isotopic.(b) For a contact structure (cid:24) on Y there is a compatible open book decomposition

of Y . Two open book decompositions compatible with a (cid:12)xed contact structure admit common

positive stabilizations.

For a complete proof of the above theorem the reader can refer to [1].

2.3 Criteria for overtwistedness

Let (cid:6) be a compact connected oriented surface with boundary. De(cid:12)ne the mapping class

group of (cid:6) to be the isotopy classes of orientation-preserving self-diffeomorphisms of the

surface (cid:6) which restrict to the identity on @(cid:6) and denote it by M CG((cid:6); @(cid:6)).

In [4],

Honda-Kazez-Matic introduced the notion of right-veering dif- feomorphisms and the monoid
V eer((cid:6); @(cid:6)) (cid:26) M CG((cid:6); @(cid:6)) of right-veering diffeomorphisms of (cid:6). We recall these notions.

8

De(cid:12)nition 2.3.1. Let (cid:11) and (cid:12) be two properly embedded arcs with a common initial
point x 2 @(cid:6). Isotope (cid:11) and (cid:12) (cid:12)xing the endpoints so that they intersect transversely with

the least possible number of points and that they are transverse to @(cid:6). We say that "(cid:12) is

to the right of (cid:11)" if (cid:11) = (cid:12) or the tangent vectors ((cid:12)

′

′

; (cid:11)

) give the orientation of (cid:6) at x.

De(cid:12)nition 2.3.2. A diffeomorphism h : (cid:6) ! (cid:6) is called right-veering if for every choice
of basepoint x 2 @(cid:6) and every choice of properly embedded arc (cid:11) based at x, h((cid:11)) is to

the right of (cid:11) at x. It is easy to see that for two isotopic self diffeomorphisms h1 and h2 of

(cid:6), h1 is right-veering if and only if h2 is right-veering. Therefore, one can talk about right

veering mapping classes. The subset of M CG((cid:6); @(cid:6)) consisting of right-veering elements is

denoted by V eer((cid:6); @(cid:6)). It follows that V eer((cid:6); @(cid:6)) is a monoid. In [4], it was shown that
the monoid Dehn+((cid:6); @(cid:6)) (cid:26) M CG((cid:6); @(cid:6)) consisting of products of right Dehn twists is a

submonoid of V eer((cid:6); @(cid:6)). The main result of [4] is the following theorem:

Theorem 2.3.3. A contact structure (M; (cid:24)) is tight if and only if all of its compatible

open book decompositions ((cid:6); h) have right-veering monodromy.

Therefore in order to prove a contact structure is overtwisted, we only have to (cid:12)nd a

compatible open book for which the monodromy is not right-veering.

Another criterion is given by by Goodman in [11] to detect overtwistedness of a contact

structure. We call an open book decomposition overtwisted if the contact structure compat-
ible with this open book is overtwisted. Let (cid:11); (cid:12) (cid:26) (cid:6) be properly embedded oriented arcs

which intersect transversely on an oriented surface F . The algebraic intersection number

ialg((cid:11); (cid:12)) is the oriented sum over interior intersections. The geometric intersection number

igeom((cid:11); ϕ((cid:11))) is the count of interior intersections regardless of sign, minimized over all

boundary (cid:12)xing isotopies of (cid:11) and (cid:12). The boundary intersection number i@((cid:11); (cid:12)) is half

of the oriented sum over the boundary intersections after minimizing interior intersections

9

(cid:12)xing the boundary.

De(cid:12)nition 2.3.4. A properly embedded arc (cid:11) (cid:26) (cid:6) is called a sobering arc for a mon-
odromy ϕ, if ialg((cid:11); ϕ((cid:11))) + igeom((cid:11); ϕ((cid:11))) + i@((cid:11); ϕ((cid:11))) (cid:20) 0, and (cid:11) is not isotopic to ϕ((cid:11)).
In particular, since i@ (cid:21) (cid:0)1 and each positive intersection contributes twice to the sum of

intersection numbers, there can be no interior intersections with positive sign. Therefore we

can reinterpret the de(cid:12)nition as follows: an arc (cid:11) is sobering if and only if, after minimizing

geometric intersections, i@ (cid:20) 0, there are no positive (internal) intersections of (cid:11) with ϕ((cid:11)),

and (cid:11) is not isotopic to ϕ((cid:11)).

The importance of sobering arcs is in the following theorem (refer to [11]):
Theorem 2.3.5. If there is a sobering arc (cid:11) (cid:26) (cid:6) for ϕ, then the open book ((cid:6); ϕ) is

overtwisted.

Example 2.3.6. The open book decomposition (S3; h) induced by negative Hopf link

H(cid:0) with (cid:12)ber surface F(cid:0). The arc (cid:11) in (cid:12)gure 2:3 is a sobering arc for the monodromy h
which is a left-handed Dehn twist. We observe that i@((cid:11); h((cid:11))) = (cid:0)1 and ialg((cid:11); h((cid:11))) =

igeom((cid:11); h((cid:11))) = 0. Therefore the induced open book is overtwisted.

h((cid:11))

(cid:11)

Figure 2.3: A sobering arc

10

2.4 Contact structure induced by the branched cover

For a transverse link L (cid:26) (S3; (cid:24)st), the 3-manifold Y obtained by the p-fold branched cover

over L can be equipped with a natural contact structure (cid:24)L. Roughly speaking, this contact

structure is obtained by lifting the standard contact structure on the knot complement to its

p-fold cover and extending it to a neighborhood of the branch set L upstairs. We describe

this construction in some detail.

Let L be a transverse knot in (S3; (cid:24)st) (if L is a link we treat each component sep-

arately).Using Darboux theorem for transverse knots, a neighborhood of L embeds into
R2 (cid:2) S1 via the coordinates (r; (cid:18); z), where (r; (cid:18)) are polar coordinates on R2, z 2 S1 and

L = r = 0 and the contact structure can be given as kernel of dz +r2d(cid:18). In this neighborhood
the covering map p : Y ! S3 is given by p((w; z)) = (wp; z)(w = rei(cid:18)). Let (cid:24)p = dz + pr2pd(cid:18)

be the kernel of the pull-back form. But this 1-form fails to be a contact form along L. To

resolve this issue, we de(cid:12)ne a new contact form by interpolating between the form dz + r2d(cid:18)

and the pull-back form in a small tubular neighborhood of L. Let ϵ1; ϵ2 < r where r is the

radius of the neighborhood above and ϵ2

1 < pϵ2p. Now set (cid:24)L = dz + f (r)d(cid:18) where f (r) = r2

for r < ϵ1 and f (r) = pr2p for r > ϵ2 and f

(r) > 0 in between. It is clear that (cid:24)L is a

′

contact form. It turns out this contact structure is independent of the choices. The reader

is referred to [3] for the details.

We can also describe the contact structure (cid:24)L on Y via open books. We represent L as a

braid of index n which intersects a generic page of the trivial open book for S3 at n points.

Then the generic page of the open book compatible with (cid:24)L will be a surface which is the

p-fold cover of the disk branched over n points. To determine the monodromy, we need to

determine how the half-twist generators of the braid L lift to the branched cover. For details

11

of this construction we refer the reader to [3]. We only describe the monodromy of this open

book.

If L has a (transverse) braid representation as (cid:27) = (cid:27)i1

:::(cid:27)ik

2 Bn so that (cid:27)ij

are some

standard generators of the braid group Bn, then the contact manifold (Y; (cid:24)L) is compatible

with an open book ((cid:6); ϕ). Here (cid:6) is the Seifert surface for the (n; p)-torus link. The lift ^(cid:27)i
of (cid:27)i 2 Bn is ti

j is a Dehn twist about the curve (cid:11)i

j as in the (cid:12)gure below

1:::ti

p(cid:0)1, which ti

(for n = p = 4). The monodromy of the open book is ϕ = (t

i1
p(cid:0)1).

ik
1 :::t

ik
i1
p(cid:0)1):::(t
1 :::t

(cid:11)1
1

(cid:11)1
2

(cid:11)1
3

(cid:11)2
1

(cid:11)2
2

(cid:11)2
3

(cid:11)3
1

(cid:11)3
2

(cid:11)3
3

Figure 2.4: Seifert surface of a (4; 4)-torus link

2.5 Homotopy invariants of contact structures

A contact structure (cid:24) regarded as an oriented 2-plane (cid:12)eld on a 3-manifold Y induces a

spinc structure which we denote by t(cid:24). Let p : (cid:25)0((cid:4)(Y )) ! spinc(Y ) be the map associating

t(cid:24) to (cid:24). We brie(cid:13)y review the classi(cid:12)cation of oriented 2-plane (cid:12)elds on Y . By trivializing
T Y and considering the oriented normal of a plane (cid:12)eld, we associate a map Y ! S2 to

(cid:24). For the case of Y = S3 the oriented 2-plane (cid:12)elds are in one-to-one correspondence with
elements of [S3; S2] = (cid:25)3(S2) = Z. By the Pontryagin-Thom construction, the space [Y; S2]

12

can be identi(cid:12)ed with framed cobordism classes of framed 1-manifolds in Y . Homotopies

outside a disk (in other words spinc structures) can be parametrized by 1-manifolds in Y
up to cobordism which corresponds to elements of H1(Y ; Z). Note there is a [S3; S2] = Z
(cid:0)1(t) for a spinc structure t, by twisting by n the given framing of the

action on the (cid:12)ber p

framed link corresponding to the oriented 2-plane (cid:12)eld. We can also see this action from

a different point of view. Consider oriented 2-plane (cid:12)elds (or the corresponding orthogonal

vector (cid:12)elds) inducing a speci(cid:12)ed spinc structure t to be identical outside a disk in Y . Then
(cid:0)1(t) by connect summing (Y; v) (v is a nonzero vector (cid:12)eld on Y ) with (S3; w),
Z acts on p

where w is a nonzero vector (cid:12)eld on S3.

By pulling back the generator of H2(S2; Z) by the map f(cid:24) : Y ! S2 associated to
(cid:24) 2 (cid:4)(Y ) we get a second cohomology class (cid:0)(cid:24) 2 H2(Y ; Z). This shows there is also a
H2(Y ; Z)-action on spinc(Y ). Now regarding (cid:24) as a complex line bundle we have c1((cid:24)) =
(cid:3)
(cid:24) (c1(T S2)) which show that c1((cid:24)) = 2(cid:0)(cid:24). Therefore as long as H2(Y; Z) has no 2-torsion,
c1((cid:24)) determines the spinc structure t(cid:24) of (cid:24).

f

Therefore the homotopy type of a 2-plane (cid:12)eld is determined by the induced spinc struc-

ture and the framing of the corresponding 1-manifold in Y . This latter invariant is generally

hard to work with except in the case of torsion c1(t(cid:24)). In this case the set of framings can
1(X; J)(cid:0) 3(cid:27)(X)(cid:0) 2(cid:31)(X)), where (X; J) is an
be lifted to Q and is calculated as d3((cid:24)) =
(c2

1
4

almost complex manifold such that @X = Y and (cid:24) is homotopic to the oriented 2-plane (cid:12)eld

of complex tangencies along @X. (cid:27)(X) and (cid:31)(X) are signature and Euler characteristic of

the manifold X respectively. The rational number d3 is called the 3-dimensional invariant

of (cid:24).

For more discussion and proofs of the above results the reader can refer to [2], chapter 6.

Now we show how to calculate the homotopy invariants from a contact surgery diagram

13

for a contact 3-manifold.

De(cid:12)nition 2.5.1.

Let K be a Legendrian knot in a contact manifold (Y; (cid:24)). By a

contact r-surgery on (Y; (cid:24)) along K we mean an r-surgery on K such that the framing is

measured with respect to the contact framing. It can be shown that the surgered manifold

Yr(K) also admits a contact structure naturally. Refer to [2], section 11.2. for more details.
We will only deal with ((cid:6)1)-contact surgery on Legendrian knots (i.e. the topological
framing of the surgery is tb(K) (cid:6) 1). According to the following theorem, every contact

manifold admits a contact surgery diagram in (S3; (cid:24)st):

Theorem 2.5.2. For any closed contact manifold (Y; (cid:24)) there is a Legendrian link L =
with framing ((cid:6)1) with respect to the

in (S3; (cid:24)st) such that contact surgery on L

(cid:0)
L+ [ L

(cid:6)

contact framings provides (Y; (cid:24)).

Refer to [5] for a proof of this theorem.

Recall that two oriented 2-plane (cid:12)elds (cid:24)1 and (cid:24)2 on a 3-manifold M are homotopic

if and only if their induced spinc structures t(cid:24)i
and 3-dimensional invariants d3((cid:24)i) are
equal. When c1(t(cid:24)) is torsion, the d3 invariant can be lifted to Q and can be computed
1(Xi; Ji) (cid:0) 3(cid:27)(Xi) (cid:0) 2(cid:31)(Xi)), where (Xi; Ji) are almost complex 4-manifolds
as d3((cid:24)i) = 1
with @Xi = M such that 2-plane (cid:12)elds of complex tangencies of Ji are homotopic to (cid:24)i along
(cid:0)
@Xi. Now we explain how to obtain the almost complex manifold X. Suppose L = L+ [ L

4(c2

is the surgery diagram for (M; (cid:24)) and let X

′

be the 4-manifold de(cid:12)ned by the diagram. X

′

admits an achiral Lefschetz (cid:12)bration (refer to [2], section 10:2: for the proof). We consider

the 2-plane (cid:12)eld of tangents to the (cid:12)bers away from the critical points. By taking orthogonal

complement with respect to some metric, we can de(cid:12)ne an almost complex structure J on

′ (cid:0) C by counterclockwise 90 degree rotation on these planes. This complex structure
X
extends to critical points corresponding to ((cid:0)1) surgeries and can be extended to points

14

corresponding to (+1)-surgeries by connect-summing with CP 2. For more details refer to
#qCP 2 (q is the number of components of L+) with extended almost

[2]. Therefore X = X

′

complex structure is our choice of (X; J) for (M; (cid:24)).

Theorem 2.5.3. The (cid:12)rst Chern class c1(X; J) 2 H2(X; Z) of the almost complex

structure discussed above evaluates on the surgery curve K as a homology class as it rotation

number: c1(K) = rot(K).

Refer to [2], chapter 11 for a proof of the above theorem.
Theorem 2.5.4. Suppose that the contact 3-manifold (Y; (cid:24)) is given by contact ((cid:6)1)-
(cid:0) (cid:26) (S3; (cid:24)st). Let X1 denote the 4-manifold de(cid:12)ned by
surgery along the link L = L+ [ L
the diagram and suppose c 2 H2(X; Z) is given by c([(cid:6)K ]) = rot(K) on [(cid:6)K ] 2 H2(X1; Z),
where (cid:6)K is the surface corresponding to the surgery curve K (cid:26) L. If the restriction cj@X1
4(c2(cid:0)3(cid:27)(X1)(cid:0)2(cid:31)(X1))+q.
the boundary is torsion and L+ has q components then: d3((cid:24)) = 1
Proof. The formula is a direct result of the above discussion and noting that (cid:31)(X1) =
(cid:31)(X1(cid:0)fx1; :::; xqg) for the critical points fx1; :::; xqg of the achiral Lefschetz (cid:12)bration X1 !
D2 which lie on the incorrectly oriented charts.

to

2.6 Concave (cid:12)llings of contact manifolds

De(cid:12)nition 2.6.1. A vector (cid:12)eld V on a symplectic manifold (W; !) is a symplectic dilation

or a Liouville vector (cid:12)eld if LV ! = 0. We say that a compact symplectic manifold (W; !) is

a convex (cid:12)lling of closed contact manifold (M; (cid:24)) if @W = M as oriented manifolds and there

exists a Liouville vector (cid:12)eld V de(cid:12)ned in a neighborhood of M , pointing out of W along

M and satisfying (cid:24) = ker((cid:19)V !jM ). In this case (M; (cid:24)) is said to be the convex boundary of

(W; !). On the other hand if V points into W along M , then we say that (W; !) is a concave

15

(cid:12)lling of (M; (cid:24)).

We have the following important fact about concave (cid:12)llings:

Theorem 2.6.2. Every contact manifold admits a concave (cid:12)lling.

Refer to [6] for a proof of this theorem.

De(cid:12)nition 2.6.3. Let the symplectic manifold (X; !) be the convex (cid:12)lling of the con-

tact manifold (Y; (cid:24)). By a convex-to-concave 2-handle H we mean a 2-handle attached
= X [ H)

symplectically to @X = Y (i.e. the symplectic structure on X extends to X

′

along a transverse knot K in (Y; (cid:24)) so that X

′

is the concave (cid:12)lling of the new boundary

′

= @(X [ H).

@X

The existence of convex-to-concave 2-handles was proved in [7]. More speci(cid:12)cally, they

show that if K is a transverse boundary component of an open book for (Y; (cid:24)) as convex

boundary of (X; !), then a 2-handle attached along K with a framing greater than the page

framing of K (the framing induced on K as boundary of page of the open book), is in fact a

convex-to-concave 2-handle. We will discuss these 2-handles in the next section to describe

concave (cid:12)llings constructed in [8] for speci(cid:12)c plumbed 3-manifolds.

2.6.1 Concave (cid:12)lling of positive plumbings

We brie(cid:13)y review the construction in [8] of the concave (cid:12)llings for speci(cid:12)c class of plumbed

3-manifolds.

De(cid:12)nition 2.6.1.1. Suppose (X; !) is a symplectic 4manifold. By a symplectic con-

(cid:12)guration in a symplectic 4-manifold we mean a union C = S1 [ :::Sn of closed symplectic

surfaces embedded in (X; !) such that all intersections between surfaces are !-orthogonal.

A symplectic con(cid:12)guration graph is a labeled graph G with no edges from a vertex to itself

and with each vertex vi labeled with a tuple (gi; mi), where gi is the genus of the symplectic

16

surface Si associated to the vertex vi and mi is the self-intersection of the surface Si. A

symplectic con(cid:12)guration graph is called positive if mi + di > 0, where di is the degree of the

vertex vi. An example is given by the graph on the top of (cid:12)gure 2:6.

The goal of the next theorem is to explicitly construct a symplectic structure !(G)

on a neighborhood N (G) of a positive symplectic con(cid:12)guration graph handle-by-handle.

Then by the sympelectic neighborhood theorem, it follows that there is a neighborhood of

any positive symplectic con(cid:12)guration ((cid:23)(C); !) in a symplectic manifold (X; !) which is

symplectomorphic to (N (G); !(G)). Moreover, the open book compatible with the contact

structure on the boundary induced by the concave (cid:12)lling will be determined.

Theorem 2.6.1.2. Let C = S1 [ :::Sn (cid:26) (X; !) be a positive con(cid:12)guration of symplectic
surfaces. Then there is a sympelectomorphism f : ((cid:23)(C); !) ! (N (G); !(G)), where G is

the con(cid:12)guration graph associated to C and (N (G); !(G)) is constructed as in the proof of

the theorem.

Proof. We only give a sketch here. For more details the reader can refer to [8]. We

explain our construction by looking at the example in (cid:12)gure 2:5. We begin with disks and

positive Hopf links as pages of open books for different copies of (S3; (cid:24)st) (cid:26) (B4; !st). The

disks are used in order to construct individual surfaces upon them and the Hopf links to

construct plumbing of the surfaces. Then we attach (4-dimensional) 1-handles with feet on

boundary of these disks or Hopf links to different copies of (S3; (cid:24)st) which contain them,

as in the bottom of (cid:12)gure 2:5. Then in the same manner we attach extra 1-handles if

necessary to raise the genus of the surfaces. These 1-handles are attached to the right

hand side of disks. To this end each surface has one boundary component. To increase

the number of boundary components if necessary, we attached more 1-handles to the left

of the disks. We note that up until this point, we have constructed a convex neighborhood

17

of the surfaces (by attaching 1-handles). Before attaching 1-handles to the left, we have

an open book for (S3; (cid:24)st) corresponding to each vertex with pf (K) = cf (K) (cid:0) di. After
attaching 2g 1-handles to to the right, still the we have pf (K) = cf (K)(cid:0) di. After attaching
mi + di (cid:0) 1 1-handles on the lower left, the surface inside will have one more component

for each handle attached, which satisfy: pf (K) = cf (K). Now attach 2-handles to each

binding component with framing pf (K) + 1. This step will turn convex (cid:12)lling to concave

ones. We will have closed surfaces so that the self-intersection of each component Fi is
(pf (K)+1)(cid:0)cf (K) = (mi +di(cid:0)1)(1)+(1(cid:0)di) = mi. This (cid:12)nishes the construction
(cid:6)K2@Fi
of the concave (cid:12)lling. By the symplectic neighborhood theorem, a neighborhood of a positive

con(cid:12)guration in a symplectic manifold should be symplectomorphic to (N (G); !(G)).

For more details of the proof refer to [8].

(0; 1)

(1; 0)

(0; 0)

Figure 2.5: construction of concave (cid:12)lling

18

2.7 Corks

De(cid:12)nition 2.7.1. A cork is a pair (W; f ), where W is a compact contractible Stein manifold
and f : @W ! @W is an involution which extends to a self-homeomorphism of W but it
does not extend to a self-diffeomorphism of W . We say that W is a cork of X if W (cid:26) X

and cutting W out of X and re-gluing it by f produces an exotic copy X

′

of X (a smooth

manifold homeomorphic but not diffeomorphic to X). This means that we have the following

decomposition: X = Y [id W and X = Y [f W , where Y = X (cid:0) int(W ).

It can be shown that any exotic copy X

′

of a closed simply-connected 4-manifold X differs

from its original copy by a cork (refer to [12], chapter 10 for more information). Figure 2:6

shows a family of corks Wn, where the involution f is de(cid:12)ned as the zero and dot exchange
on their underlying symmetric links. For instance, W1 is a cork of E(2)#C (cid:22)P 2 (refer to [12]

and references therein for further discussion). We will encounter the cork W1 in section

where we look at a speci(cid:12)c contact structure on its boundary.

0

n

n + 1

f

Figure 2.6: The family of corks Wn

19

Chapter 3

Main results

3.1 Concave (cid:12)llings

We recall the de(cid:12)nition of a positive plumbing from last chapter. Consider a plumbing of

closed symplectic surfaces P = S1 [ :::Sn in a symplectic manifold (X; !). To this Plumbing

we associate a plumbing graph consisting of a vertex for each symplectic surface and an

edge between two vertices if the two symplectic surfaces are plumbed together. Let mi be

the self-intersection and di the degree of the vertex vi. This plumbing is called positive if

mi + di > 0 for each vertex vi. In [8] it was proved that such a plumbing of symplectic

surfaces has a neighborhood that is a concave (cid:12)lling of its boundary and the induced contact

structure on the boundary is compatible with an open book as follows: the generic page is a

surface obtained by connect-summing the surfaces Si as in the plumbing con(cid:12)guration and

there are mi + di boundary components for each surface Si. The monodromy consists of

one positive Dehn twist about each boundary curve and one negative Dehn twist about each

neck of the connect-sum.

We demonstrate this with an example. Look the plumbing graph of symplectic surfaces

in (cid:12)gure 3:1, where (gi; mi) for each vertex vi means the surface Si has genus gi and self-

intersection equal to mi. Then induced contact structure on its boundary is compatible with

the following open book with monodromy ϕ = (cid:5)(cid:14)i(cid:5)(cid:27)

(cid:0)1
j where (cid:14)i are Dehn twists about blue

20

curves (about boundary components) and (cid:27)j are Dehn twists about the red curves (about

the neck) as in the (cid:12)gure 3:1.

(0; 1)

(1; 0)

(0; 0)

Figure 3.1: a concave plumbing graph and its corresponding open book

In this section we show that in some cases a ((cid:0)1)-framed symplectic sphere in a positive

plumbing can be blown down to obtain another positive plumbing with the same induced

contact structure on the boundary. We then present an algorithm to construct a positive

concave (cid:12)lling for each Brieskorn manifold (but the concave (cid:12)lling we obtain is not unique).

Then we look at an example (cid:6)(2; 3; 5) and draw the contact surgery diagram for the contact

structure induced by this concave (cid:12)lling. We compute homotopy invariants of this contact

structure and compare them to those of the standard Milnor (cid:12)llable one. We then show that

any concave (cid:12)lling of a Brieskorn manifold constructed by the above algorithm, induces an

overtwisted contact structure on it.

Theorem 3.1.1. Suppose we have a positive co(cid:12)guration of symplectic surfaces as in
(cid:12)gure 3:2. By blowing down the middle ((cid:0)1)-sphere, we obtain another positive con(cid:12)guration

with the same contact boundary (i.e. both con(cid:12)gurations are concave (cid:12)llings of the same

contact 3-manifold).

21

(m; g)

((cid:0)1; 0)

′

)

(n; g

Figure 3.2: A special positive plumbing graph

Proof. According to the lantern relation, we have tatbtctd = t(cid:11)t(cid:12)t(cid:13) as in (cid:12)gure 3:3 below:

a

(cid:12)

(cid:11)

b

c

d

(cid:13)

Figure 3.3: the lantern relation

Now we consider a 4-holed sphere as (cid:12)gure 3:4 below. This type of picture of a 4-holed

sphere will be useful for our argument. From the lantern relation we obtain the relation

(cid:0)1
(cid:0)1
(cid:11) = t
c

(cid:0)1
d t(cid:12)t(cid:13). The left and right pictures in (cid:12)gure 3:4 correspond to the curves
t

tatbt

involved in left and right hand sides of this relation. By destabilizing the monodromy

(removing a positive Hopf band) we obtain the last picture in the (cid:12)gure:

(cid:11)

a

b

(cid:13)

c

(cid:12)

d

a

(cid:12)

b

Figure 3.4: removing a positive Hopf band

Now we prove the theorem by considering a speci(cid:12)c example as in the (cid:12)gure 3:5 where

m = 2, n = 1 and g = g

′

= 0, but our argument applies to the general case as well.

22

The induced open book on the boundary has a page as in the top picture in the (cid:12)gure

3:5 and the monodromy is ϕ = t(cid:14)1

t(cid:14)2

t(cid:14)3

(cid:0)1
t
(cid:27)1

t(cid:14)4

(cid:0)1
t
(cid:27)2

t(cid:14)5

t(cid:14)6

. Now if we trace the pictures in the

(cid:12)gure backwards, we conclude that this open book is equivalent to the bottom picture as

′
the page and monodromy ϕ

= t(cid:14)1

t(cid:14)2

t(cid:14)3

t(cid:14)4

(cid:0)1
t
(cid:27)1

t(cid:14)5

t(cid:14)6

t(cid:14)7

.

2

(cid:0)1

1

(cid:14)1

(cid:14)2

(cid:14)3

(cid:27)1
(cid:14)4

(cid:27)2
(cid:14)5

(cid:14)6

3

2

(cid:14)1

(cid:14)2

(cid:14)3

(cid:14)4

(cid:27)1

(cid:14)5

(cid:14)6

(cid:14)7

Figure 3.5: Blowing down and the corresponding open books

It is easy to see that this new open book corresponds to the concave plumbing graph

below, which proves the theorem in our special case. The general case is similar.

Next we show that all Brieskorn manifolds are boundary of positive concave plumbings.

Lemma 3.1.2. Each Brieskorn manifold (cid:6)(p; q; r) admits a positive concave plumbing.

Proof. We know that a Brieskorn manifold (cid:6)(p; q; r) is a Seifert (cid:12)bered space with three

singular (cid:12)bers and the base a genus zero surface (refer to [13], theorem 2:1: for a proof).

Thus it has a plumbing diagram as in (cid:12)gure 3:6. If for the central vertex mi < (cid:0)2 or if there
is a middle vertex with mi < (cid:0)1 or an end vertex with mi < 0, we blow up a +1-sphere

between this vertex and a vertex next to it. This will increase increase self-intersection of the

two old vertices by one and the new +1-sphere already satis(cid:12)es the condition mi + di > 0 (in

this case 1 + 2 > 0). It is easy to check that after a (cid:12)nite steps this will give us a plumbing

graph as desired. We demonstrate this with and example.

Example 3.1.3. (cid:6)(2; 3; 5) has the following surgery description: 15b1 + 10b2 + 6b3 = 1

23

Figure 3.6: plumbing diagram of a Brieskorn manifold

with b1 = (cid:0)1, b2 = 1 and b3 = 1 (refer to [14], theorem 6:7: for obtaining surgery diagram

of a Brieskron manifold).

We apply the above algorithm to (cid:12)nd a concave plumbing graph from this surgery dia-

gram. The result is shown in (cid:12)gure 3:7.

5

0

1

2

1

5

(cid:0)2

0

3

Figure 3.7: A positive plumbing graph for (cid:6)(2; 3; 5)

3

According to [8] the concave plumbing graph on the right induces the following open book

on its boundary. The red curves correspond to negative Dehn twists and the blue curves to

positive Dehn twists. In order to be able to realize the surgery curves as Legendrian curves

we present this surface as in (cid:12)gure 3:8.

Figure 3.8: the open book on (cid:6)(2; 3; 5)

We notice that the (cid:12)rst (cid:12)ve Dehn twists from the top can be removed. The (cid:12)rst two

24

Figure 3.9: A Legendrian diagram for (cid:6)(2; 3; 5)

since they are positive and negative twists about the same curve and the next three positive

twists since they correspond to positive stabilizations. Then the surgery diagram of the

contact structure compatible with this open book will be given as in (cid:12)gure 3:9. Since we

perform negative Dehn twists about all the red curves, the framing on each surgery curve is

a (+1)-contact framing:

Now we calculate the homotopy invariants of the this contact structure (cid:24). Since Y =

(cid:6)(2; 3; 5) is a homology sphere, the (cid:12)rst obstruction d2((cid:24)) 2 H2(Y ; Z) = 0. Now recall that
(c2 (cid:0) 3(cid:27)(X) (cid:0) 2(cid:31)(X)) + q, where X is the handlebody
the next obstruction is d3((cid:24)) =
obtained by attaching 2(cid:0)handles to D4 along the surgery curves, q is the number of +1-
surgery curves, c 2 H2(X; Z) is given by c([(cid:6)i]) = rot(ki) on [(cid:6)i] 2 H2(X; Z) where (cid:6)i is

1
4

the Seifert surface corresponding to a component ki of the diagram. Finally (cid:27)(X) and (cid:31)(X)

are the signature and Euler characteristic of X.

The linking matrix is as follows:

25

26666666664

(cid:0)17 (cid:0)14 (cid:0)6 (cid:0)4
(cid:0)14 (cid:0)13 (cid:0)6 (cid:0)4
(cid:0)6 (cid:0)6 (cid:0)5
(cid:0)4 (cid:0)4

0
0 (cid:0)3

37777777775

We compute (cid:27)(X) = (cid:0)2 and (cid:31)(X) = 5. Now we look at the long exact sequence of the

pair (X; @X): 0 ! H2(@X; Z) ! H2(X; Z) !ϕ1 H2(X; @X; Z) !ϕ2 H1(@X; Z) ! 0.

The maps ϕ1 and ϕ2 are calculated as follows: ϕ1([(cid:6)i]) = (cid:6)lk(ki; kj)[Nj] and ϕ2([Ni]) =

[(cid:22)i] (Ni is a disc bounding the meridian (cid:22)i). Now P D(c) = (cid:6)rot(ki)[Ni] = 17[N1]+13[N2]+
5[N3]+3[N4]. We (cid:12)nd the solution to ϕ1(C) = P D(c) as C = 17[(cid:6)1](cid:0)47[(cid:6)2]+37[(cid:6)3]+41[(cid:6)4]
(14 (cid:0) 3((cid:0)2) (cid:0)
(for negative choice of rotation numbers). Thus c2 = C2 = 14 and d3((cid:24)) =

1
4

2:5) + 4 =

13
2

.

We can compare the result to homotopy invariants of the standard contact structure (cid:24)st
on (cid:6)(2; 3; 5) induced by its Milnor (cid:12)ber (cid:0)E8. Let the complex polynomial F : C3 ! C be
given by F (x; y; z) = x2 + y3 + z5 and U be a connected open subset of C3 containing the
origin. If w0 2 F (U ) is a regular value, then the compact smooth manifold with boundary
(cid:0)1(w0) \ B6 is the Milnor (cid:12)ber (cid:8) = (cid:8)(2; 3; 5) of the Brieskorn manifold (cid:6)(2; 3; 5).
can be shown that (cid:8) has a plumbing description as in (cid:12)gure 3:10 called the (cid:0)E8 plumbing

It

F

(refer to [12], chapter 12 for further discussion). As the Milnor (cid:12)ber (cid:8) is the pre-image
of a regular value under F , its normal bundle (cid:23)(cid:8) is trivial, as is T C3j(cid:8). Since (cid:23)(cid:8) :=
T C3j(cid:8)
, we conclude the tangent bundle T (cid:8) is trivial and therefore c1((cid:8)) = c1((cid:24)st) = 0.
T (cid:8)
By looking at the plumbing (cid:0)E8, we conclude that (cid:31)((cid:8)) = 9 and (cid:27)((cid:8)) = (cid:0)8. Thus

d3((cid:24)st) =

1
4

(0 (cid:0) 2:9 (cid:0) 3((cid:0)8)) =

3
2

are not homotopic.

. Therefore we conclude that these two contact structures

26

(cid:0)2 (cid:0)2 (cid:0)2 (cid:0)2 (cid:0)2 (cid:0)2 (cid:0)2

(cid:0)2

Figure 3.10: The (cid:0)E8 plumbing

In fact we can prove that the contact structure induced by the concave (cid:12)lling is over-

twisted. (The contact structure induced by (cid:0)E8 is tight since (cid:0)E8 is a Stein (cid:12)lling of the

Poincare sphere.)

As mentioned above, there are many positive concave plumbings that (cid:12)ll a (cid:12)xed Brieskorn

sphere. We prove that the contact structure induced on a Brieskorn sphere, by a positive

concave plumbing according to the above algorithm, is always overtwisted (i.e. it does not

matter which plumbing we choose). We prove this by using the sobering arc technique which

was mentioned in section 2:3:.

Theorem 3.1.4. A positive concave plumbing constructed by the above algorithm, always

induces an overtwisted contact structure on a Brieskorn sphere.

Proof. We (cid:12)rst construct a concave (cid:12)lling of Brieskorn sphere as in lemma 3:2. We then

construct the open book as described at the beginning of this section. We modify the open

book using the move in (cid:12)gure 3:5. We start with one of the end vertices v1 and remove

all but one of the original boundary components corresponding to this vertex. Again we

demonstrate with an example. We can use a similar argument for a general graph. Suppose

one branch of the plumbing ends with vertices as follows:

2

1

Figure 3.11: A branch in the plumbing

Then the corresponding open book has one end as below and we apply the move in

(cid:12)gure 3.4 to remove its boundary components one by one:

27

(a)

(b)

(c)

(d)

Figure 3.12: modifying the open book

The grey curves on each surface mean that there will be a negative Dehn twist about the

curve in the next picture. Look at (cid:12)gure 3.12 (c). We observe that the negative and positive

Dehn twists about the (cid:12)rst boundary component from right cancel each other. Thus we get

(cid:12)gure 3.12 (d) where there is no Dehn twist about this component. In the same (cid:12)gure we

have sketched an arc (cid:11) from this boundary component to another one, together with its

image under the monodromy. We can check that the arc (cid:11) is a sobering arc: All the interior

intersections are negative and and the boundary intersection is equal to zero. Therefore the

contact structure compatible with this open book is overtwisted.

3.2 A contact structure on the boundary of the cork

The cork W1 can be seen as double branched cover over a ribbon disk (refer to [12], chapter

11 for further discussion). Its boundary M = @W1 is the double branched cover over the

ribbon knot K as shown in (cid:12)gure 3:13.

In this section we examine boundary of the cork as the contact manifold (M; (cid:24)) that arises

as the double branched cover over K realized as a transverse knot in (S3; (cid:24)st). We construct

the open book compatible with this contact structure using the techniques discussed in

28

section 2:4 and we use the right-veering criterion to prove that it is overtwisted.

0

2 : 1

branched cover

Figure 3.13: Cork as branched cover over a ribbon disk

Theorem 3.2.1. The ribbon knot K has the following presentation as a braid in B4:

(cid:27) = (cid:27)

(cid:0)1
3 (cid:27)2(cid:27)

(cid:0)2
1 (cid:27)

(cid:0)1
2 (cid:27)3(cid:27)1(cid:27)

(cid:0)1
2 (cid:27)

(cid:0)1
1 (cid:27)2(cid:27)3(cid:27)

(cid:0)1
(cid:0)1
2 (cid:27)2
3(cid:27)
2

Let (@W1; (cid:24)) be boundary of the cork equipped with the contact structure (cid:24) induced by the

double branched cover over the transverse knot K corresponding to the braid (cid:27). Then (cid:24) is

overtwisted.

Proof. Using the techniques discussed in section 2:4: we construct an open book com-

patible with the contact structure (cid:24). Then using the right-veering criterion, we prove that

this contact structure in overtwisted. Recall from section 2:4: the contact structure (cid:24) is

supported by an open book ((cid:6); ϕ) such that a generic page (cid:6) is equal to the Seifert surface

of a (2; 4)-torus link (p = 2 and n = 4 n this case) which is a twice-punctured torus, and its

monodromy is given as below:

ϕ = t

(cid:0)1
b

(cid:0)1
b

t2
ct

(cid:0)1
(cid:0)1
a t
b

tctbt

(cid:0)1
b

(cid:0)1
(cid:0)2
a tbt
t
c

tatct

Refer to (cid:12)gure 3:14 for the surface (cid:6) and the curves a, b and c on it. For our convenience

we rewrite this monodromy as follows:

(cid:0)1
ϕ = (t
c
(cid:0)1
(cid:0)1
= t
t
c

tct
t

(b)

(cid:0)1
t
b
(cid:0)1
b

(cid:0)1
tc)tc(t
b
(cid:0)1
(cid:0)1
t
a (b)
t

(c)

tctb)(t
(cid:0)2
(cid:0)1
t
b

tct

(a)

(cid:0)1
(cid:0)1
a t
b

(cid:0)1
ta)tc(t
b
(cid:0)1
t
b

(cid:0)2
a tb)
t
(cid:0)1
(cid:0)1
t
a (b)
t

(c)

= tct

(cid:0)2
(cid:0)1
tct
t
b

(cid:0)1
(cid:0)1
t
t
c

(b)

(a)

29

 ((cid:11))

a

b

(cid:11)

c

Figure 3.14: (cid:11) and  ((cid:11))

Now if ϕ 2 V eer((cid:6); @(cid:6)), composing it with t

t

(cid:0)1
c

(b)

(cid:0)1
t2
t
b

(a)

would give us another right-

veering diffeomorphism (since V eer((cid:6); @(cid:6)) is a monoid according to the previous section).

We show that   = ϕot

(cid:0)1
c

t

(b)

(cid:0)1
t2
t
b

(a)

= tct
t

(cid:0)1
b

(c)

(cid:0)1
(cid:0)1
t
a (b)

t

tc is not right-veering. Therefore ϕ

cannot be right-veering either and (M; (cid:24)) is overtwisted. Figure below shows an arc (cid:11) and

its image  ((cid:11)) on the surface (cid:6). It is clear that   is not right-veering for this arc. Therefore

the result follows.

Remark 3.2.2. It might be possible to check directly that ϕ is not right-veering, by

examining its effect on some arc in (cid:6). But since ϕ contains a lot of words, we decided to

proceed as above as a shortcut.

30

BIBLIOGRAPHY

31

BIBLIOGRAPHY

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32