KNOT THEORY OF MORSE-BOTT CRITICAL LOCI

By

Metin Ozsarfati

A DISSERTATION

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Mathematics—Doctor of Philosophy

2018

ABSTRACT

KNOT THEORY OF MORSE-BOTT CRITICAL LOCI

By

Metin Ozsarfati

We give an alternative proof of that a critical knot of a Morse-Bott function f : S3 → R

is a graph knot where the critical set of f is a link in S3 [8] [9] [11] [12]. Our proof inducts

on the number of index-1 critical knots of f as in [12].

Copyright by
METIN OZSARFATI
2018

ACKNOWLEDGMENTS

I, Metin Ozsarfati, would like to take this opportunity to thank to my supporters who

have made important contributions to this thesis. First and foremost, to my thesis advisor

Matthew Hedden who was supportive of my independent study or goals and was patient

during my preparation of an earlier draft of this thesis. He gave valuable comments on this

thesis and provided me important references for further study. A certain part of the proof

of Lemma 6 is due to him. I still emotianally felt his guidance and his sincere aﬀection even

during the times when we saw each other rarely. He had been an active teacher about a

variety of topology classes at MSU, especially those related to low dimensional topology or

knot theory for which I am grateful. Secondly, I thank to the referee of “Topology and its

Applications” journal for bringing the important paper [12] to my attention. Lastly, this

thesis has been supported by “Douglas A. Spragg Endowed Fellowship Award, 2017” and

“MSU College of Natural Science Dissertation Completion Fellowship, 2018”.

iv

TABLE OF CONTENTS

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

vi

Chapter 1

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

Chapter 2 Basic Notions and Results

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

Chapter 3 Round Handle Decomposition of an ordered k-function . . . .

1

3

9

Chapter 4 Characterization of k-mate knots . . . . . . . . . . . . . . . . . .

15

Chapter 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

v

LIST OF FIGURES

Figure 2.1: The ﬁgure shows some of the trajectories of a gradient like vector ﬁeld for
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

f1.

Figure 2.2: The situation for K1#K2
Figure 3.1: A {θ} × D2 cross section of a k-model neighborhood of K1

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

5

8

. . . . . . .

11

vi

Chapter 1

Introduction

We will deﬁne and study a speciﬁc type of Morse-Bott functions [1] f : S3 → R, called a

k-function in this thesis, to study knots or links in S3 where each critical component of f

is a knot in S3. A knot or link in S3 is called k-mate if it is a sublink of the critical link

of some k-function. Our main theorem (Theorem 1) will classify all the k-mate knots in S3

which are precisely the graph knots [11] in S3. Our treatment will be similar to the study of

an ordinary Morse function on a compact manifold and we will adopt the standard notions

of Morse theory [13] in our work .

Unfortunately, there are many diﬀerent but equivalent deﬁnitions of a graph knot or link

in a compact 3-manifold M . Our deﬁnition in this work will give one equivalent deﬁnition

for a graph knot in S3 but it will slightly fall short to capture all the graph links in S3.

A standard deﬁnition can be taken as: an irreducible link L in an irreducible, compact,

connected 3-manifold M is an irreducible graph link if the JSJ-decomposition [2], [3] of
M −N (L) consists of only Seifert ﬁbered pieces where N (L) is an open tubular neighborhood

of L in M [11]. Such 3-manifolds are called graph manifolds [4]. Another characterization
of this link L is that the Gromov volume of M − N (L) is zero [5], [6], [7]. For compact,

connected M with ∂M being equal to a (possibly empty) union of tori or klein bottles,
another charactirization is that L ⊆ M is a graph link if L is a subset of the hyperbolic

closed orbits of a nonsingular Morse-Smale ﬂow on M [9]. There, Morgan’s main interest

1

had actually been the existence of nonsingular Morse-Smale ﬂows on M rather than studying

links in M but he proved that when M is prime, there exists such a ﬂow on M if and only if

M is a graph manifold. For M not necessarily prime, he proved that there exists such a ﬂow

on M if and only if each prime summand of M is a graph manifold. Such ﬂows on M are

bijectively associatied to round handle decompositions of M [9], [10]. Here, a round handle

is either homeomorphic to a solid torus or a solid Klein bottle. We will describe in our work

how a k-function induces a round handle decomposition of S3.

It is shown in [8], [11], [12] that a graph knot K is obtained from an unknot by a ﬁnite

application of connected sum or cabling operations and we will show that all the k-mate knots

in S3 arise in the same way. We will formally emphasize this elementary perspective of graph

knots in Deﬁnition 3. In literature, [9] seems to be the earliest source to be credited for the

classiﬁcation of graph knots even though the connected sum operation has been overlooked

there.

Even though our main theorem will classify the graph knots in a diﬀerent way, our

topological ideas and methods will be close to the ones in [12]. The results in [11], [12]

are stronger than ours as they eﬀectively classify all the graph links in S3. Moreover, the

classiﬁcation in [11] studies graph links in a homology 3-sphere M . We will make further

remarks on these important sources in the Conclusion section.

2

Chapter 2

Basic Notions and Results

Deﬁnition 1. A real valued smooth function f : S3 → R is a k-function if:

(i) The set of critical points of f is a link L in S3

(ii) The Hessian of f is nondegenerate in the normal direction to L.

(iii) Each knot K in L has a tubular neighborhood U in S3 with local coordinates (θ, x, y)
such that f (θ, x, y) = c2(±x2 ± y2) + d where (θ, 0, 0) are the coordinates for K and
c, d are scalars (c (cid:54)= 0).

The link L is called the critical link of f and a component of L is called a critical knot of

f . The neighborhood U in (iii) is called a k-model neighborhood of K. We make an abuse
of notation by identifying U ⊆ S3 with S1 × D2 where D2 is the unit disk. The notation

(θ, x, y) will refer to such local coordinates of a critical knot throughout this work.

Deﬁnition 2. A link L is k-mate if L is a subset of the critical link of some k-function.

The basic question is then which knots in S3 are k-mate. Our main theorem below

answers this question. We will provide a proof of it after Theorem 9.

Theorem 1. A knot is k-mate if and only if it is a graph knot.

A critical knot K of a k-function f is called a source, sink or saddle respectively if the
signs in f (θ, x, y) = c2(±x2±y2)+d are both positive, both negative or opposite respectively.
We adopt the sign conventions in f (θ, x, y) = c2(y2 − x2) + d for a saddle.

3

In the saddle K case, the two circles S1 × (±1, 0) are called stable circles of K in U and
the two circles S1 × (0,±1) are called the unstable circles of K in U . Similarly, the annulus
S1 × [−1, 1] × {0} is called the stable annulus of K in U and the annulus S1 × {0} × [−1, 1]

is called the unstable annulus of K in U . Note that neither k-model coordinates (θ, x, y)

nor stable or unstable circles of a saddle are unique. The stable and unstable circles of a

saddle K can be isotoped to K within the stable or unstable annulus so that they are parallel
cable knots of K. They homologically have ±1 longitude coeﬃcients (and some arbitrary

meridian coeﬃcient) in H1(∂U ). Note that the property (iii) of Deﬁnition 1 is not necessary

for sources or sinks but it puts a restriction on our saddles. When it is dropped, the stable

or unstable regions of a saddle can be a nontrivial line bundle over the saddle (i.e. a M¨obius

band).

A point in f (L) ⊆ R where L is the critical link of a k-function f is called a critical
value of f and a point in R − f (L) is called a regular value of f . We will deﬁne an ordered

k-function later and Lemma 6 shows that the preimage of a regular value of an ordered

k-function is a collection of disjoint tori in S3.

Given a k-function f , there exists a gradient like vector ﬁeld X on S3 for f . More
∂x ±
∂y ) around a critical knot of f (see e.g. [14] for the existence of a gradient like vector

precisely, Xp(f ) is positive if p is not a critical point of f and also, X(θ, x, y) = c2· (±2x ∂
2y ∂

ﬁeld for a Morse function). The function f is increasing on the forward ﬂow lines of X.
For any point p in S3 , the ﬂow line Xt(p) converges to a critical point of f as t → ±∞.

An important application of X is that the ﬂow of X gives an isotopy between the regions
f−1((−∞, r]) and f−1((−∞, r + ]) in S3 (here,  > 0) when f−1([r, r + ]) does not contain

any critical points of f .

A source of a k-function is a sink of −f and vice versa. We may always assume a k-mate

4

knot to be a source of some k-function by the following lemma.

Lemma 2. A knot K is a source of some k-function if and only if it is a saddle of some

k-function.

Proof. We will prove only one direction as the other one is similar. Suppose that K is a

saddle of a k-function f . Let ˜U be a k-model neighborhood of K. Let D be the disk of radius
1/2 centered at the origin in R2 and consider the smaller k-model neighborhood U = S1× D

of K inside ˜U .

Consider the isotopic knots K1 := S1 × (−2/3, 0) and K2 := S1 × (−5/6, 0) and take
a small tubular neighborhood Vi of Ki in Int( ˜U ) so that Vi intersects each meridian disk
{θ} × D2 of ˜U in a disk (See Fig. 2.1). Moreover, the intersections V1 ∩ U = ∂V1 ∩ ∂U and
V1∩V2 = ∂V1∩∂V2 are both annuli the core of which are isotopic to K and also, V2∩U = ∅.
We can deﬁne a k-function f1 by modifying f only within ˜U − U so that V1 contains a k-

model neighborhood of the source K1 of f1 and V2 becomes a k-model neighborhood of the

saddle K2 of f1. Here, K is isotopic to the source K1.

Figure 2.1: The ﬁgure shows some of the trajectories of a gradient like vector ﬁeld for f1.

Lemma 3. Let f be a k-function without any saddles. Then, f has a single source and a

single sink which form a Hopf link in S3.

5

Proof. Since S3 is closed, f has at least one source K1 and a sink K2. Let X be a gradient

like vector ﬁeld for f and Xt be the ﬂow of X. Take a point p in ∂U1 where U1 is a k-model

neighborhood of K1. The point Xt(p) will be in a k-model neighborhood U3 of a sink K3 of
f for large enough t since f has no saddles. Say, Xap(p) ∈ U3 for some ap > 0. Since ∂U1 is
compact and connected and f does not have any saddles, we have Xa(∂U1) ⊆ U3 for some
time a ≥ ap. We may assume that Xa(∂U1) = ∂U3 after scaling X with a positive smooth
function on S3 if necessary. Then, Xa(U1) ∪ U3 is an embedded, closed and connected 3-
manifold in the closed and connected S3. Therefore, Xa(U1) ∪ U3 is S3. Hence, the source

K1 and the sink K3 = K2 are the only critical knots of f . Let P1 and P3 denote the core
of the solid tori of Xa(U1) and U3 respectively. The union Xa(U1) ∪ U3 gives a lens space
description of S3 so that the two solid tori Xa(U1) and U3 are two complementary standard
solid tori in S3 by the topological classiﬁcation of lens spaces. Therefore, P1 ∪ P3 (cid:39) K1 ∪ K3

is a Hopf link.

The above lemma shows that an unknot is k-mate. The next two lemmas describe a way

to construct other k-mate knots and as we will show later in Theorem 1, all k-mate knots

arise in this way starting with the unknot.

A knot K is a cable knot of J if K can be isotoped into ∂U where U is a closed tubular

neighborhood of J in S3. Here, K is allowed to bound a disk in ∂U so that an unknot is a

trivial cable knot of any knot J. Even when K does not bound a disk in ∂U so that K is

not a trivial cable knot of J, the cable knot K can be a meridian of J or a longitude of an
unknot J so that K is still a trivial knot. We will use the notation K (cid:39) Jp,q which says that

the cable knot K of J is homologically p longitudes plus q meridians of J. We will sometimes
conveniently suppress the coeﬃcients p and q and use the notation K (cid:31) J instead.

6

Lemma 4. A cable knot K of a k-mate knot J is k-mate.

Proof. If K is trivial, then it is k-mate by Lemma 1. Otherwise, K can be isotoped to be

transverse to the meridian disks of k-model neighborhood U of J. We may assume that

J is a source of a k-function f by Lemma 2. The rest of the proof will follow exactly as

in that lemma where f gets modiﬁed only within U but still preserving a smaller tubular

neighborhood of J. The knot K becomes another source and a saddle isotopic to K gets

inserted between K and J.

A connected sum K1#K2 of two knots K1 and K2 is not well deﬁned in general unless

both K1 and K2 and their ambient spaces S3’s are all oriented. One can regard K1 and

K2 as a split link in the same ambient space S3 and an orientation of this single S3 can

be ﬁxed easily. However, a k-function does not induce a natural orientation on a critical

knot of it. While we study k-functions, we will strictly work with unoriented knots. The
notation K1#K2 will then denote a knot in the set {K+
K±

2 } of knots where

speciﬁes an orientation of Ki.

i

1 #K+

2 , K+

1 #K−

Lemma 5. A connected sum K1#K2 of k-mate knots K1 and K2 is k-mate.

Proof. The k-mate knots K1 and K2 are sources of some k-functions f1 and f2 by Lemma

2 respectively. Let S be a sphere in S3 which yields the connected summands K1 and K2 of
K1#K2. Let ˜S be a small closed tubular neighborhood of S in S3 so that ˜S ∩ K1#K2 is two
unknotted arcs in ˜S (cid:39) S2 × [0, 1]. Let V be a small closed tubular neighborhood of K1#K2
in S3 such that V ∩ ∂ ˜S is four disjoint disks and also, V ∪ ˜S is smoothly embedded in S3.
Let J denote the core of the annulus S − V . The region S3 − Int(V ∪ ˜S) has two connected

components each of which is diﬀeomorphic to the complement of K1 or K2 in S3. Let Ri
denote the component of S3 − Int(V ∪ ˜S) that is diﬀeomorphic to the complement of Ki.

7

Let Ui be a small closed k-model neighborhood of the source Ki of fi. We may assume that
S3 − Int(Ui) = Ri by isotoping Ui in S3. See Figure 2.2. By using the ﬂow of the gradient

like vector ﬁelds for f1 and f2, we may scale f1 and f2 and add some constants so that they

agree in a small tubular neighborhood of ∂U1 or ∂U2 with f1(∂U1) = f2(∂U2) = 1 (see e.g.

[14] for such scaling of fi).

We can now deﬁne a k-function f such that:

(i) V is a k-model neighborhood of the source K1#K2 of f with f (K1#K2) = 0

(ii) ˜S − V contains a k-model neighborhood of the saddle J of f with f (J) = 0.5 and also,

S − V is a stable annulus of J in ˜S − V .

(iii) f|Ui

= fi|Ui

Therefore, K1#K2 is k-mate.

Figure 2.2: The situation for K1#K2

8

Chapter 3

Round Handle Decomposition of an

ordered k-function

Before studying the preimage of a regular value of f and how it changes when we pass a

critical level, we ﬁrst introduce ordered k-functions where we make local modiﬁcations near

the critical link L of f without changing the critical set of f or the type of each component

of L. The number  > 0 will denote a suﬃciently small positive number throughout the text.

For a source K of f with local k-model coordinates (θ, x, y), we can use an increasing
smooth function h : [0, 1] → (−∞, 0] with h(z) = 0 near z = 1 and linear near z = 0 to
change f locally by redeﬁning ˜f (θ, x, y) := f (θ, x, y) + h(x2 + y2) ≤ f (θ, x, y) so that f

can have arbitrarily small values on the source K. Similarly, f can be redeﬁned near a sink

to have an arbitrarily large value on it. For a saddle K of f , we can use a decreasing (or
increasing) smooth h : [0, 1] → [0, ] (or [−, 0]) and h(z) = ± near z = 0 and h(z) = 0 near
z = ±. We can then redeﬁne f near K as ˜f (θ, x, y) = f (θ, x, y) + h(x2 + y2) which changes
the saddle value f (K) by ±. Here, |h(cid:48)(z)| is also small enough so that K remains to be a

saddle of ˜f without creating any other critical points.

An ordered k-function f has then the following properties:

(i) The critical values of the critical knots of f are all distinct.

(ii) The critical values of f are ordered as: source values ≤ saddle values ≤ sink values.

9

Say, a1 < ··· < aj < b1 ··· < bk < c1 < ··· < cm where ai, bi and ci correspond to a

source, a saddle and a sink of a ordered k-function f respectively. Recall the smallness of :
if z0 is a critical value of f , then z0 is the only critical value of f in [z0 − , z0 + ]. We now
describe a round handle decomposition [9] of S3 by analyzing the preimages of an ordered

k-function f . Such an analysis will be repeatedly used in our proofs.

Start with r < a1 having f−1((−∞, r]) = ∅. When we increase r, each time r passes a
source value of f , the preimage f−1([a1, r]) will have one more solid torus in S3; a round
0-handle is attached to the empty set. The region f−1([a1, ai + ]) will consist of i disjoint

solid tori.

When we pass b1, the preimage ˜V := f−1([a1, b1 +]) is the union of V := f−1([a1, b1−])
which consists of j disjoint solid tori and a region f−1([b1 − , b1 + ]). One connected

component R1 of this latter region is a solid torus that contains a k-model neighborhood of

the saddle K1 where Ki is the saddle of f with f (Ki) = bi. The component R1 contains
a tubular neighborhood ˜A1 of the stable annulus of K1 in f−1([b1 − , b1 + ]). For an
appropriate choice of ˜A1, one can show that f−1([a1, b1 + ]) is isotopic to V ∪ ˜A1 in S3 (see
e.g. [13] for a Morse analogue of this fact). This tubular neighborhood ˜A1 is attached to V
along two disjoint annuli in ∂V ∩ ∂ ˜A1 the cores of which are isotopic to the unstable circles
of K. The region f−1([a1, b1 + ]) is topologically equivalent to the union of f−1([a1, b1 − ])
and a solid torus ˜A1 in S3 that intersect each other along two parallel annuli in ∂ ˜A1 ∩ V
the cores of which have ±1 longitude coeﬃcients in H1(∂ ˜A1). In this case, ˜A1 is a round

1-handle that is attached to V along two annuli that are tubular neighborhoods of stable

circles in ∂V .

The consecutive passes of saddle values of f look similar. Each time we pass a saddle

value bi, the region f−1([a1, bi + ]) is isotopic to the union V := f−1([a1, bi − ]) and a

10

Figure 3.1: A {θ} × D2 cross section of a k-model neighborhood of K1

solid torus ˜Ai in S3 where V ∩ ˜Ai = ∂V ∩ ∂ ˜Ai consists of two parallel annuli the cores
of which have ±1 longitude coeﬃcients in H1(∂ ˜Ai). Here, the solid torus ˜Ai is a tubular

neighborhood of a stable annulus of the saddle Ki. When we pass bi, the boundary of the
preimage changes from ∂V to ∂(V ∪ ˜Ai) by a surgery on the two stable circles C1 and C2 of
Ki in ∂V ∩ ∂ ˜Ai. Speciﬁcally, a tubular neighborhood (cid:39) S1 × ∂I × I of C1 and C2 in ∂V gets
replaced by another two disjoint annuli (cid:39) S1 × I × ∂I which is now a tubular neighborhood
of the unstable circles of Ki in ∂(V ∪ ˜A1). They have the following identiﬁcation where
I = [−1, 1]:

(cid:39)−→ S1 × (I − {0}) × ∂I

S1 × ∂I × (I − {0})
(θ, x, ty) −→ (θ, tx, y) where t ∈ (0, 1] and x, y = ±1
We will use the notation s(·) to denote a surgered surface in S3 coming from the pass of
a saddle value of f so that in the above situation, the surface s(∂V ) is isotopic to ∂(V ∪ ˜Ai)
in S3.

When we pass the ﬁrst sink value c1, a k-model neighborhood U of the sink in f−1(c1)
ﬁlls in V := f−1([a1, c1 − ]) in S3. The region f−1([a1, c1 + ]) is isotopic to V ∪ U in S3
where V ∩ U = ∂V ∩ ∂U is a torus. Here, U is a round 2-handle that is attached to V along

11

a torus. So, the boundary of f−1([a1, c1 − ]) is m disjoint tori and each time we pass a sink

value, one of those m tori is ﬁlled in by a solid torus coming from a k-model neighborhood
of a sink. This process ends with f−1([a1, cm]) = S3.

As we have explained above, an ordered k-function on S3 induces a round handle de-

composition of S3. The converse induction almsot holds. The only exceptions where this

induction fails depend on the round 1-handles. In our deﬁnition of a saddle, the stable and

unstable annuli of a saddle are trivial line bundles over S1. In the round 1-handles deﬁned

in [9], such unstable and stable line bundles over S1 may be non-trivial; i.e. they can be

M¨obius bands. Our restriction on saddles implies that every k-mate link is a graph link but

not necessarily the converse. However, we will still be able to provide an alternative proof

to Theorem 1 which states that a knot is k-mate if and only if it is a graph knot. Lemma 2

shows that a knot can be realized as a saddle of a k-function if and only if it can be realized

as a source of a k-function and our deﬁnition of a source is general without any restrictions.

We brieﬂy remark that each ordered k-function naturally deﬁnes a Morse function on

S3 with the following indices of critical points: A pair of 0 and 1 from a source, a pair of
1 and 2 from a saddle and a pair of 2 and 3 from sink. For each such a {j, j + 1} critical

index pair, the attaching sphere of the j + 1-handle intersects the belt sphere of the j-handle

geometrically twice.

The converse also holds. If f0 : S3 → R is a Morse function such that:

(i) The critical values of f0 are all distinct.

(ii) The indices of the critical points of f0 come in adjacent pairs and their critical values
are ordered on the real line R as: First {0, 1} pairs, then {1, 2} pairs and then {2, 3}

pairs

12

(iii) For each {j, j + 1} index pair of critical points pj and pj+1 of f0 respectively, the

attaching sphere of the j + 1-handle of pj+1 intersects the belt sphere of the j-handle
of pj geometrically twice (also, with the same signs when {j, j + 1} = {1, 2}).

Then, f0 induces a ordered k-function f on S3. Each {0, 1},{1, 2} and {2, 3} index pair

of paired critical points of f0 gives rise to a source, saddle and a sink of f respectively.

An alternative proof of the below lemma is in [9].

Lemma 6. If f is an ordered k-function and r is a regular value of f , then each connected
component of f−1(r) is an embedded torus in S3.

Proof. Let a1 < ··· < aj < b1 ··· < bk < c1 < ··· < cm denote the critical values of f where

ai, bi and ci correspond to a source, a saddle and a sink of f respectively. The lemma is
clear for r < b1 or r > bk. Assume now that for some b1 < r < bk, the surface f−1(r) has a

non-torus component. Since a torus has Euler characteristic 0 and the Euler characteristic

of f−1(bi − ) does not change after a surgery during the pass of bi, there exists a sphere ˆS1
in some f−1(w1) where w1 is a regular value of f .

The sphere ˆS1 bounds a 3-ball on each side in S3 and let B1 denote the one of them such
that B1 ∩ f−1(w1 − ) = ∅. As f−1(z) is a union of tori for a regular value z with z > bk,
there must be a surgery on a sphere S1 isotopic to ˆS1 in B1 during the pass of a saddle

value but it may happen that the surface s(S1) produced by surgery contains a sphere in
B1. Take a regular value w2 ≥ w1 large enough such that f−1(w2) contains a sphere S2 in

B1 but the produced surface s(S2) does not contain a sphere after the pass of a saddle value

β2. Moreover, we can ﬁnd such w2 and S2 such that the 3-ball B2 bounded by S2 in S3 with
B2 ∩ f−1(w2 − ) = ∅ satisﬁes B2 ⊆ B1.

Let K2 be the saddle with f (K2) = β2. Let A2 denote the stable annulus of K2 and

13

{C1, C2} = ∂A2 denote the stable circles of K2 with C1 ⊆ S2. Then, C2 is not in S2 but
in another component Σ of f−1(β2 − ) with genus greater than 1 because s(S2) does not

contain a sphere.

The existence of such Σ with big genus implies that B2 contains at least one source K and
B2∩ f−1(b1− ) (cid:54)= ∅. Moreover, a surgery must happen on (not necessarily distinct) tori Ta

and Tb in B2 containing the stable circles Ca and Cb of a saddle respectively, such that both

Ca and Cb bound disjoint disks in Ta and Tb respectively. This surgery then produces also
a sphere S3. Say, S3 ⊆ f−1(w3). Let B3 denote the 3-ball bounded by S3 in S3 such that
B3 ∩ f−1(w3 − ) = ∅. We can ﬁnd such S3 and B3 such that B3 ⊆ B2 and K ⊆ B2 − B3.

We can now apply our last argument to S3 = ∂B3 instead of S1 = ∂B1 to conclude that B3
contains at least one source J with J (cid:54)= K, B4 ⊆ B3 and J ⊆ B3 − B4 where B4 is a 3-ball
in S3 and the sphere ∂B4 is in the preimage of a regular value of f . Therefore, B1 contains

inﬁnitely many sources of f and we have reached the desired contradiction.

14

Chapter 4

Characterization of k-mate knots

Theorem 9 will essentially state the equivalence between k-mate knots and graph knots. Its

proof will be divided into several cases most of which will be handled locally. These local

arguments are not diﬃcult when one studies each possible situation with a careful thought.

There is an exceptional case (the Subcase 3 of Case 3) though where a local analysis does

not suﬃce as in the previous cases and we will employ various technical methods to tackle

this diﬃcult case. We ﬁrst prove the below technical lemma which will provide us a nonlocal

picture in S3 in certain situations. We will later deﬁne a graph kit of a graph knot (Deﬁnition

4) and strengthen the statement of Theorem 9 with parts (i), (ii) and (iii) there to obtain a

more global picture in this exceptional case. We will be able to complete the proof of this

case with these extra technical details at hand.

Suppose that K1 and K2 are two unknots such that K1 is a cable knot of K2. Since

both K1 and K2 are unknots, K2 is then a cable knot of K1 as well. If it is a trivial cabling
or K2 is a longitude of K1, then K1 ∪ K2 is a split link of two unknots. If lk(K1, K2) = ±1,
then K1 ∪ K2 is a Hopf link. The below lemma exposes such cabled two unknots K1 and

K2 in generality but we will encounter many split links of two unknots or Hopf links in its

proof.

Lemma 7. Suppose that K is a critical unknot of an ordered k-function f and KR is the
unknot core of an unknotted solid torus R in S3 such that K ∩ R = ∅ and also, ∂R is in

15

the preimage of a regular value of f . Then, the unknots K and KR are cable knots of each

other.

Proof. If f has no saddles, then Lemma 3 shows that K ∪ KR is a Hopf link. Assume now
that f has a saddle and let a1 < ··· < aj < b1 < ··· < bk < c1 < ··· < cm denote the critical

values of f where ai, bi and ci correspond to a source, a saddle and a sink of f respectively.

Let K1 be the saddle of f with f (K1) = b1. We will induct on the number of saddles k by
analyzing the stable circles C1 and C2 of K1 in f−1(b1 − ) and also a stable annulus A of
K1 in f−1([b1 − , b1]). We may assume r /∈ [b1 − , b1 + ] where ∂R ⊆ f−1(r). The circles

C1 and C2 are cable knots of (not necessarily distinct) sources P1 and P2 respectively. Let
Ei denote the solid torus component of f−1([a1, b1 − ]) containing Pi. Let E denote the
component of f−1([a1, b1 + ]) which contains P1 ∪ P2 ∪ K1. In each case below, we will
deﬁne an ordered k-function f1 with at most k − 1 saddles.
Case 1. Only one of C1 and C2 bounds a disk in f−1(b1 − ).

Say, C1 bounds a disk D in f−1(b1 − ) so that K1 (cid:39) C1 is an unknot. Then, P2 cannot

be nontrivial. Otherwise, C2 must be a meridian of P2 and P2 will intersect the sphere,

which is the union of D, A and a meridian disk of P2, geometrically once. As H2(S3) = 0,

such a single geometric intersection of a 1-cycle and a 2-cycle of S3 is not possible. So, P2 is
an unknot and C2 is a longitude of P2 as it bounds the disk D ∪ A in the complement of P2.
We consider the situation P1 (cid:54)= P2 ﬁrst. The region E is then isotopic to E1 in S3. If
K is equal to K1 or P2, then K is contained in a small 3-ball B containing the disk A ∪ D
such that B ∩ KR = ∅. The link K ∪ KR is then a split link of two unknots.

Assume now that K is distinct from K1 and P2. We deﬁne a k-function f1 by f1(p) :=
f (p) for p /∈ E so that E becomes a k-model neighborhood of the source P1 of f1. Then, f1 is

16

ordered because the region f−1((−∞, b1 +]) contains just a single saddle of f . If E∩R = ∅,
then r is a regular value of f1 with ∂R ⊆ f−1

1 (r). An application of the induction hypothesis

to the critical unknot K of f1 and the unknotted solid torus R proves the lemma.

If R ⊆ E, then R is a tubular neighborhood of P2 or an unknot P1. If R is a tubular
neighborhood of P2, then KR (cid:39) P2 is contained in a 3-ball not containing K and K ∪ KR is
a split link of two unknots. If R is a tubular neighborhood of an unknot P1, then K (cid:54)= P1.
Also, R is isotopic to E in S3 and K (cid:42) E as K is distinct from K1 and P2. So, the induction

hypothesis applies to the critical unknot K of f1 and the unknotted solid torus E to prove

the lemma in this situation.

We now prove the lemma for the situation P1 = P2. Then, E is a solid torus and P1∪ K1

is a split link of two unknots. If K is equal to P1 or K1, then K is inside a 3-ball B in
E such that B ∩ KR = ∅ so that K ∪ KR is a split link of two unknots. Assume now
K (cid:42) E. Let KE denote the core of E. We deﬁne f1 by f1(p) := f (p) for p /∈ E so that
E becomes a k-model neighborhood of the source KE of f1. If E ∩ R = ∅, we can apply
the induction hypothesis as before. If R ⊆ E, then R is either a k-model neighborhood of

P1 or KE is an unknot and both R and E are tubular neighborhoods of KE. In the former
case, KR is contained in a 3-ball within E not containing K so that K ∪ KR is a split link

of two unknots. In the latter case, we apply the induction hypothesis just as before where
K ∪ KE (cid:39) K ∪ KR.
Case 2. Both C1 and C2 bound disks D1 and D2 in f−1(b1 − ) respectively.

Then, K1 is an unknot saddle. We ﬁrst consider the case that the sources P1 and P2

are distinct.

In this situation, there are two possible ways to attach the round handle

corresponding to K1 to the solid tori E1 and E2. One way leads to a sphere component of

17

∂E which is not possible by Lemma 6. We must have the other possibility where ∂E is a
disjoint union of two tori. The sphere D1 ∪ D2 ∪ A bounds 3-ball B1 and B2 on either side
in S3 and the cut of E along this sphere produces two punctured solid tori in S3.

If the interior of some Bi contains only one of the knots K and KR, then K ∪ KR is a
split link of two unknots. Otherwise, say K ∪ KR ⊆ Int(B1). The 3-ball B2 contains only
one of P1 and P2. Say, P2 ⊆ B2 and P1 ∩ B2 = ∅. The region E ∪ B2 is then isotopic to
E1 in S3. We deﬁne f1 by f1(p) := f (p) for p /∈ E ∪ B2 so that E ∪ B2 becomes a k-model

neighborhood of the source P1 of f1. An application of the induction hypothesis ﬁnishes the
proof when R (cid:42) E. When R is a subset of E − B2, it is then a tubular neighborhood of P1
and it is isotopic to E ∪ B2 in S3. In this situation, we apply the induction hypothesis for
the unknotted solid torus E ∪ B2 and the critical knot K of f1.

Assume now that P1 = P2. The disks D1 and D2 cannot be disjoint since otherwise,
f−1(b1 + ) would contain a sphere contradicting Lemma 6. Say, D2 ⊆ D1. Let A1 denote
the annulus D1 − Int(D2). The torus T0 = A1 ∪ A separates S3 into two closed regions and
let R0 denote the one of them such that Int(R0) ∩ E1 = ∅. Similarly, let ˜R0 denote the the
component of S3 − Int(E) such that ˜R0 is isotopic to R0 in S3.

As R0 in S3 is bounded by a torus, it is diﬀeomorphic to the complement of a knot K0
in S3 (after smoothing the corners of T0). Take a small 3-ball identiﬁed with D2 × [0, 1]

coming from the push oﬀ of the disk D2 into the exterior of E1 in a normal direction so
that D2 × {0} := D2 ⊆ ∂E1 and ∂D2 × [0, 1] ⊆ A. We can ﬁrst take K0 to be the union

of a properly embedded arc in A1 and another properly embedded arc in A. We can then

slightly push oﬀ this union of two arcs into the exterior of R0 in a normal direction to achieve
K0 ∩ R0 = ∅. Then, S3 − R0 is a tubular neighborhood of K0 and D2 × {0} is a meridian
disk of K0. Therefore, B0 := R0 ∪ D2 × [0, 1] is diﬀeomorphic to a 3-ball which intersects

18

E1 in the disk D1. Hence, the region E1 ∪ R0 ∪ D2 × [0, 1] is isotopic to E1 in S3 and
similarly, so is the region E ∪ ˜R0. So, the surface s(∂E1), which has two components, has
one component isotopic to ∂E1 and another component isotopic to T0 in S3.

If K = K1, then K can be isotoped along A into the 3-ball D2 × [0, 1] and we can easily
isotope KR out of this 3-ball if necessary without removing K from that 3-ball. So, K ∪ KR
is a split link of two unknots. We will assume K (cid:54)= K1 from now on.

If (K ∪ R) ∩ (E ∪ ˜R0) = ∅, we deﬁne f1 by f1(p) := f (p) for p /∈ E ∪ ˜R0 so that E ∪ ˜R0

becomes a k-model neighborhood of the source P1 of f1. An application of the induction

hypothesis to the critical unknot K of f1 and the solid torus R proves the lemma.

If (K ∪ R) ⊆ ˜R0, we deﬁne f1 by f1(p) := f (p) for p ∈ ˜R0 so that S3 − ˜R0 becomes a

k-model neighborhood of the source K0 of f1. We can then apply the induction hypothesis

just as before.

Assume now that only one of R and K is inside ˜R0 and the other is outside ˜R0. Say,
Ka ⊆ ˜R0 where {Ka, Kb} = {KR, K}. Then, Ka is contained in the 3-ball B0 but Kb is not
so that K ∪ KR is a split link of two unknots.

The ﬁnal possible situation is that only one of R and K is inside E but none of them are

inside ˜R0. Then, P1 is either equal to K or isotopic to KR within R and in the latter case, we
may assume P1 = KR. Say, Ka = P1 where {Ka, Kb} = {KR, K}. Then, Kb (cid:42) E ∪ ˜R0. We
deﬁne f1 by f1(p) := f (p) for p /∈ E ∪ ˜R0 so that E ∪ ˜R0 becomes a k-model neighborhood

of the source P1 of f1. An application of the induction hypothesis proves the lemma.
Case 3. None of C1 and C2 bounds a disk in f−1(b1 − ).

Subcase 1. Both C1 and C2 bound meridian disks D1 and D2 of P1 and P2 respectively.

Then, K1 is an unknot saddle. The sources P1 and P2 are equal since otherwise, P1

19

would intersect the sphere S1 := D1∪ D2∪ A geometrically once in S3. The sphere S1 yields
P1 (cid:39) Pa#Pb and the region S3 − E has two components Ra and Rb which are isotopic to
the complement of Pa and Pb in S3 respectively.

We consider the situation K = K1 ﬁrst. If R ⊆ E, then P1 is an unknot and R is a
tubular neighborhood of it. We see that K ∪ KR is a Hopf link in this setting. If R (cid:42) E,
then say R ⊆ Rb. Since K is in the 3-ball Ba bounded by S1 and containing the region Ra
but KR is not in Ba, the link K ∪ KR is a split link of two unknots. We will assume K (cid:54)= K1

from now on.

Assume R ⊆ E so that R is a tubular neighborhood of the unknot P1 and Pa and Pb are
unknots as well. As K (cid:54)= K1, the unknot K is either in Ra or Rb. Say, K ⊆ Ra. We deﬁne
f1 by f1(p) := f (p) for p ∈ Ra so that S3 − Ra becomes a k-model neighborhood of the

unknot source Pa of f1. We can now apply the induction hypothesis to the critical unknot

K of f1 and a k-model neighborhood of the source Pa of f1 to conclude that Pa and K are

cable knots of each other. The lemma then follows because Pa can be isotoped to KR within
S3 − Ra so that Pa ∪ K (cid:39) KR ∪ K.

If R (cid:42) E, then say R ⊆ Rb. If K ⊆ Rb, we deﬁne f1 by f1(p) := f (p) for p ∈ Rb so
that S3 − Rb becomes a k-model neighborhood of the source Pb of f1. An application of the
induction hypothesis proves the lemma. If K ⊆ Ra, then K is in the 3-ball Ba not containing
R so that K ∪ KR is a split link of two unknots. If K (cid:42) Ra ∪ Rb, then K = P1 because
K (cid:54)= K1 as well. This ﬁnal situation is similar to the previous situation where R ⊆ E and
K ⊆ Ra ∪ Rb.

Subcase 2. Only C1 bounds a meridian disk D1 of P1.

Then, K1 is an unknot saddle. The sources P1 and P2 are distinct since C2 is a non-

20

meridian, nontrivial cable knot of P2. Since C2 bounds the disk D1 ∪ A1, both C2 and P2

are unknots and C2 is a longitude of P2. Also, P2 is a meridian of P1 and E is isotopic to

E1 in S3.

We ﬁrst consider the situation where K is equal to K1 or P2. If R ⊆ E, then R is a
tubular neighborhood of either P1 or P2. If R ⊇ P2, then K ∪ KR (cid:39) C2 ∪ P2 is a split link
of two unknots. If R ⊇ P1, then P1 is an unknot and K ∪ KR (cid:39) C1 ∪ P1 is a Hopf link. If
R (cid:42) E, then K is contained in a small 3-ball B inside E with KR ∩ B = ∅ so that K ∪ KR

is a split link of two unknots. We will assume that K is distinct from K1 and P2 from now

on.

Assume R (cid:42) E. We deﬁne f1 by f1(p) := f (p) for p /∈ E so that E becomes a k-model

neighborhood of the source P1 of f1. The induction hypothesis can then be applied as before.
Assume now R ⊆ E so that R is a tubular neighborhood of either P1 or P2. If K = P1,
then KR is isotopic to P2 within R and K ∪ KR (cid:39) P1 ∪ P2 is a Hopf link. Assume now
K ∩ E = ∅. If R ⊇ P2, then KR is isotopic to P2 within R where P2 is in the 3-ball B not
containing K so that K ∪ KR (cid:39) K ∪ P2 is a split link of two unknots. If R ⊇ P1, we deﬁne
f1 by f1(p) := f (p) for p /∈ E so that E becomes a k-model neighborhood of the unknot

source P1 of f1. The induction hypothesis can be applied as before.

Subcase 3. None of C1 and C2 is a meridian of P1 and P2 respectively.

The isotopic cable knots C1 and C2 are, say, C1 (cid:39) (P1)p,q and C2 (cid:39) (P2)r,s where p, r (cid:54)= 0
as Ci is not a meridian of Pi. We will ﬁrst consider the situation P1 (cid:54)= P2. Let ˜A be a closed
tubular neighborhood of the stable annulus A so that the annulus ˜Ci := ˜A ∩ Ei becomes a
tubular neighborhood of Ci in ∂Ei and also, E is isotopic to E1∪E2∪ ˜A in S3. The boundary
of E1∪E2∪ ˜A is a torus which comes from the union of the two annuli ∂Ei−Int( ˜Ci) (i = 1, 2)

21

and the two annuli that are parallel copies of A in ˜A. Let R1 := S3 − Int(E1 ∪ E2 ∪ ˜A).

If p = ±1, then P1 is isotopic to C1 in E1. As C1 and C2 are isotopic, we get P1 (cid:31) P2
(i.e., P1 is a cable knot of P2). Similarly, if r = ±1, then P2 (cid:39) K1 and P2 (cid:31) P1. In these
situations, E is isotopic to E1 (when P2 (cid:31) P1) or E2 (when P1 (cid:31) P2) in S3.

We will now prove that P1 ∪ P2 is a Hopf link when p, r (cid:54)= 0,±1. The solid torus
E1 admits a Seifert ﬁbration with a single singular ﬁber P1 of multiplicity |p| so that the
annulus ˜C1 becomes a union of regular ﬁbers because C1 is not a meridian of P1. This

ﬁbration on ˜C1 extends to a regular Seifert ﬁbration on ˜A because C1 is a cable knot of
K1 with C1 (cid:39) (K1)±1,β. We may assume that ˜C2 ⊆ ∂ ˜A is a union of regular ﬁbers since
the cable knots C1 and C2 of K1 have the same slope. This ﬁbration on ˜C2 can then be
extended to a Seifert ﬁbration of E2 with a single singular ﬁber P2 of multiplicity |r| because
C2 is not a meridian of P2. So, the region E1 ∪ E2 ∪ ˜A becomes a Seifert ﬁbered manifold
over a disk with two singular ﬁbers of multiplicities |p| and |r|. The torus ∂(E1 ∪ E2 ∪ ˜A)
bounds E1 ∪ E2 ∪ ˜A and R1 in S3 at least one of which must be a solid torus. Since
π1(E1 ∪ E2 ∪ ˜A,∗) (cid:39)< z, w; zp = wr >(cid:54)(cid:39) Z, the region R1 must be a solid torus. A regular
ﬁber in ∂(E1 ∪ E2 ∪ ˜A) = ∂R1 is nontrivial there. It cannot bound a meridian disk in R1
since otherwise π1(S3,∗) = π1(E1 ∪ E2 ∪ ˜A ∪ R1,∗) (cid:39)< z, w; zp = wr = 1 >(cid:54)(cid:39) 1. Therefore,

the Seifert ﬁbration on ∂R1 can be extended into R1 with at most one singular ﬁber so that
we obtain a Seifert ﬁbration of S3 = E1∪E2∪ ˜A∪R1 over a sphere with two or three singular
ﬁbers. It follows now from the classiﬁcation of Seifert ﬁbered manifolds that S3 cannot have

a Seifert ﬁbration with three singular ﬁbers over a sphere but only two so that R1 must have

a regular Seifert ﬁbration (see e.g. [15] or [16]). Moreover, if one takes the base sphere as

the union of two disks each of which contains a point corresponding to a singular ﬁber P1 or

P2, then those two disks will correspond to two complementary solid tori in S3 so that the

22

cores P1 and P2 of those two complementary solid tori form a Hopf link in S3.

We ﬁrst consider the cases P1 (cid:31) P2 or P2 (cid:31) P1 where E is isotopic to E1 or E2 in S3.
Say, Pa (cid:31) Pb where {Pa, Pb} = {P1, P2}. Assume R (cid:42) E. We deﬁne f1 by f1(p) := f (p)
for p /∈ E so that E becomes a k-model neighborhood of the source Pb of f1.

If K is

distinct from Pa and K1, then an application of the induction hypothesis proves the lemma.
If K is equal to Pa or K1, then Pb is also an unknot because Pa (cid:39) K1 is a nontrivial,

non-meridian cable knot of Pb. Moreover, Pa is isotopic to Pb within E. An application of

the induction hypothesis to the critical unknot Pb of f1 and R proves the lemma because
K ∪ KR (cid:39) Pb ∪ KR.

Assume now R ⊆ E. Then, R is a tubular neighborhood of (possibly both) Pa or Pb.

In either case, Pb is an unknot because Pa is a nontrivial, non-meridian cable knot of Pb.
Also, KR is isotopic to Pb within E. If K ⊆ E, then K is equal to Pa, Pb or K1 and also,
K ∪ KR (cid:39) Pa ∪ Pb which proves the lemma. If K (cid:42) E, we deﬁne f1 by f1(p) := f (p) for
p /∈ E so that E becomes a k-model neighborhood of the source Pb of f1. We apply the

induction hypothesis just as before.

We consider the Hopf link P1 ∪ P2 case now. We still have both P1 (cid:31) P2 and P2 (cid:31) P1
but E is no longer isotopic to E1 or E2 in S3. The torus knot C1 (cid:39) (P1)p,q is nontrivial
since p (cid:54)= 0,±1. The region V := S3 − Int(E) is a solid torus the core of which is isotopic to
C1 (cid:39) K1.

Assume K ∪ R ⊆ V . Since the core of V is nontrivial, each of the unknots K and KR
is contained in some 3-balls BK and BR inside V respectively. Let g : V → S3 be an
embedding such that g(V ) is a standard, unknotted solid torus in S3. Then, both g(K) and

g(KR) are unknots since each of K and KR is contained in a 3-ball inside V . We deﬁne f1
by f1(p) := f (g−1(p)) for p ∈ g(V ) so that S3 − g(V ) becomes a k-model neighborhood of

23

an unknot source Kg of f1. The unknots g(KR) and g(K) are then cable knots of each other
by the induction hypothesis. Therefore, so is the link g−1(g(K) ∪ g(KR)) = K ∪ KR.

Assume now that only one of K and R is inside V . The one inside V is then contained
in a 3-ball not containing the other one so that K ∪ KR is a split link of two unknots. The
ﬁnal remaining case is K ∪ R ⊆ E where K ∪ KR (cid:39) P1 ∪ P2 is a Hopf link.

We will now prove this subcase of the lemma for the situation P1 = P2. Let ˜A, ˜Ci and
C1 (cid:39) (P1)p,q (p (cid:54)= 0) be just as before where C1 is now isotopic to C2 in ∂E1. The nontrivial

circles C1 and C2 separates ∂E1 into two closed annuli A1 and A2 and the components of
s(∂E1) are isotopic to the tori Σ1 := A1 ∪ A and Σ2 := A2 ∪ A in S3. Let Hi denote the
closed region bounded by Σi in S3 such that Int(Hi) ∩ E1 = ∅. Similarly, let ˜Hi denote the
component of S3 − Int(E) that is isotopic to Hi in S3.

Assume that H1 is not a solid torus. Then, E1 ∪ H2 bounded by Σ1 is a solid torus.
If C1 bounds a disk in E1 ∪ H2, then C1 is a meridian of the core of E1 ∪ H2 and also a

longitude of the unknot P1 because C1 is a nontrivial, non-meridian cable knot of P1. The
region E1 ∪ H2 is then isotopic to H2 in S3 so that H2 is a solid torus. When C1 does not
bound a disk in E1 ∪ H2, the region E1 ∪ H2 admits a Seifert ﬁbration with at most one

single singular ﬁber where the annuli A1 and A in its boundary become a union of regular

ﬁbers. As ∂A2 = ∂A1 consists of two regular ﬁbers, the annulus A2 can then be isotoped
into ∂(E1 ∪ H2) relative to its boundary in the Seifert ﬁbered solid torus E1 ∪ H2 so that
H2 is again isotopic to E1 ∪ H2 in S3.

Therefore, at least one of H1 and H2, say H1, is a solid torus. Let KH denote the core of
both H1 and ˜H1. The union E1∪H1 of two solid tori intersecting each other at an annulus in
their boundaries is then similar to the union E1 ∪ E2 ∪ ˜A in our previous situation P1 (cid:54)= P2.
Therefore, either P1 ∪ KH is a Hopf link with C1 being a nontrivial torus knot or one of the

24

knots P1 and KH is a cable knot of the other one. In the latter case, E1 ∪ H1 is isotopic to
E1 or H1 in S3. In the former Hopf link case, the region H2 is also a solid torus the core of

which is isotopic to C1.

We have S3 = E ∪ ˜H1 ∪ ˜H2 where the interiors of those three regions are disjoint. There

are various possibilities about where K and R might be. We start with the assumption
R ∪ K ⊆ ˜H2. The case where P1 ∪ KH is a Hopf link and C1 is a nontrivial torus knot has
already been analyzed in the “Hopf link P1 ∪ P2” situation before and the lemma holds in
this case. If P1 (cid:31) KH or KH (cid:31) P1 and also, E1 ∪ H1 is isotopic to E1 or H1 in S3, we
deﬁne f1 by f1(p) := f (p) for p /∈ E ∪ ˜H1 so that E ∪ ˜H1 becomes a k-model neighborhood

of the source P1 or KH of f1. An application of the induction hypothesis proves the lemma.
Assume now R∪ K ⊆ ˜H1. The case where KH is nontrivial has already been analyzed in
the “Hopf link P1∪P2” situation before and the lemma holds in this case. If KH is an unknot,

we apply the induction hypothesis more directly by simply deﬁning f1 by f1(p) := f (p) for
p ∈ ˜H1 so that S3 − ˜H1 becomes a k-model neighborhood of an unknot source of f1.

Assume now that only one of R and K is in ˜H1 and the other one is in ˜H2. Say, Ka ⊆ ˜H1
where {Ka, Kb} = {KR, K}. If KH is nontrivial so that ˜H1 is a knotted solid torus, then
K ∪ KR is a split link of two unknots. If KH ∪ P1 is a Hopf link and C1 is a nontrivial
torus knot, then ˜H2 is a knotted solid torus and K ∪ KR is again a split link of two unknots.
The remaining situation is that KH is trivial and E ∪ ˜H1 is isotopic to ˜H1 in S3. Also, ˜H2
is an unknotted solid torus in S3. Here, each of the unknots K and KR are in one of the
two complementary unknotted solid tori E ∪ ˜H1 and ˜H2 but not in the same one. Let JH
denote the unknot core of ˜H2 so that KH ∪ JH is a Hopf link. We deﬁne f1 by f1(p) := f (p)
for p ∈ ˜H1 so that E ∪ ˜H2 becomes a k-model neighborhood of the source JH of f1. An
application of the induction hypothesis shows that Ka is a cable knot of JH . Since KH ∪ JH

25

is a Hopf link, Ka is a cable knot of KH as well by rediﬁning the core KH of H1 if necessary.

A similar argument shows that the other unknot Kb is a cable knot of KH as well. Since
Ka ⊆ ˜H1 but Kb (cid:42) ˜H1, the unknot Ka can be isotoped into an arbitrarily small tubular
neighborhood of KH without aﬀecting Kb. Therefore, the unknots in Ka ∪ Kb = KR ∪ K

are cable knots of each other as well.

The only possibility we haven’t considered so far is that R or K is inside E. If K∪R ⊆ E,
then R is a tubular neighborhood of P1 and K = K1. The lemma follows from K ∪ KR (cid:39)
C1 ∪ P1 and C1 (cid:31) P1. Assume now that only one of K and R is inside E. If R ⊆ E, we
may assume KR = P1. Say, Ka ⊆ E where {Ka, Kb} = {K, KR}. Then Ka is equal to P1

or K1 and in the latter case, P1 is also an unknot because the unknot K1 is a nontrivial,

non-meridian cable knot of P1.

First assume Kb ⊆ ˜H2. If P1 ∪ KH is a Hopf link and C1 is a nontrivial torus knot, then
˜H2 is a knotted solid torus and K ∪ KR is a split link of two unknots. Otherwise, E ∪ ˜H1 is
isotopic to E1 in S3 and we deﬁne f1 by f1(p) := f (p) for p ∈ ˜H2 so that E ∪ ˜H1 becomes a
k-model neighborhood of the unknot source P1 of f1. The unknots in P1 ∪ Kb are then cable
knots of each other by the induction hypothesis. So are the unknots in K1 ∪ Kb because K1

can be isotoped into an arbitrarily small neighborhood of P1 without aﬀecting Kb. Since
KR ∪ K is either equal to P1 ∪ Kb or K1 ∪ Kb, the lemma is proven in this situation.

Assume now Kb ⊆ ˜H1. If KH is nontrivial, then K ∪ KR is a split link of two unknots.
If P1∪ KH is a Hopf link, the unknots K and KR are in two complementary unknotted solid

tori but not in the same one and the lemma has been proven in this situation above. The
remaining case is that KH is trivial and E ∪ ˜H1 is isotopic to both E1 and ˜H1 in S3 where
the unknots KH and P1 are cable knots of each other. In this case, ˜H2 is a standard solid
torus and P1 ∪ JH (cid:39) KH ∪ JH is a Hopf link where JH is the core of ˜H2. We deﬁne f1 by

26

f1(p) := f (p) for p ∈ E ∪ ˜H1 so that ˜H2 becomes a k-model neighborhood of the unknot
source JH of f1. An application of the induction hypothesis shows Kb (cid:31) JH so that Kb can
be isotoped into ∂ ˜H1 within ˜H1 and hence, Kb (cid:31) P1. We now see Kb (cid:31) Ka as well even
when Ka (cid:54)= P1 since in this case Ka = K1 and, Kb = KR can be isotoped into an arbitrarily

small tubular neighborhood of P1 without aﬀecting K1. This completes the proof of the

lemma.

We will take the following elementary description as a deﬁnition of a graph knot [8], [11].

Deﬁnition 3. Let S0 := {unknot}. To deﬁne Sn inductively (n ∈ N), assume that Sk is
deﬁned for 0 ≤ k < n and let Sn denote the set of cable knots of P where P is a connected
sum of knots in Sn−1. A knot in S := ∪k∈N Sk is called a graph knot.

We have the following facts about the set of graph knots S: The set S1 is the set of

(trivial or nontrivial) torus knots. Since the cable knot (K)1,r of any knot K is isotopic
to K, we have Sn+1 ⊇ Sn for all n ∈ N.
If {K1, . . . , Km} ⊆ Sn, then K1#··· #Km (cid:39)
(K1#··· #Km)1,r ∈ Sn+1. If we have a sequence of cable knots U ≺ K1 ≺ K2 ≺ ··· ≺ Km
where U is an unknot, then Ki ∈ Si for 1 ≤ i ≤ m.

Deﬁnition 4. Suppose that K is a graph knot in Sn. If n = 0, we deﬁne the graph knot

kit or shortly the graph kit of K to be the empty set. For n > 0, ﬁx a (not necessarily
unique) expression of K with K (cid:39) (P )q,r and P (cid:39) PK,1#··· #PK,m where PK,i ∈ Sn−1
for 1 ≤ i ≤ m. We deﬁne Γ(K) corresponding to this ﬁxed expression of K by Γ(K) :=
{PK,1, . . . , PK,m}. Let Φ1 := Γ(K). For 1 ≤ j < n, we deﬁne Φj+1 inductively by Φj+1 :=
Φj ∪ {J : J ∈ Γ(H) where H ∈ Φj} where the elements PK,i ∈ Γ(K) and PJ,s ∈ Γ(J) are

distinct elements of Φj+1 whenever K and J are distinct in Φj. Then, we say that Φn is a

graph kit of K and also, K is woven from the graph knots in Φn.

27

Note that even when the expression K (cid:39) (P )q,r and P (cid:39) PK,1#··· #PK,m of K ∈ Sn is
unique, we can consider K in Sn+1 ⊇ Sn and may produce a diﬀerent graph kit of K. Also,

distinct graph knots in a graph kit can be isotopic. For example, for the torus knots T2,3 and
T2,5 in S1, the set Φ := {T2,3, T2,5, unknot2,3, unknot2,5} is a graph kit of T2,3#T2,5 (where

Φ is valid for any orientations of T2,3 and T2,5 deﬁning T2,3#T2,5).

If Φ is a ﬁnite collection of unoriented knots, then #
J∈Φ

J denotes a connected sum of the

knots in Φ which are assigned an arbitrary orientation before their connected sum is taken.

If Φ is a graph kit of some graph knot and P is a nontrivial graph knot in Φ, then ΓΦ(P )

will denote the ﬁnite set of graph knots such that P is a cable knot of a connected sum of
the graph knots in ΓΦ(P ) and also, ΓΦ(P ) ⊆ Φ. In this case, the orientations of the knots

in the connected sum

J are not arbitrary but in such a way so that the graph knot

#

J∈ΓΦ(P )

P becomes a cable knot of the graph knot

J.

#

J∈ΓΦ(P )

Lemma 8. A graph knot K is k-mate.

Proof. Say, K ∈ Sn. If n = 0, then K is an unknot which is k-mate by Lemma 3. Assume

now that K is nontrivial and n > 0. We induct on n. Each graph knot in Γ(K) is k-mate

by the induction hypothesis. Applications of Lemma 4 and Lemma 5 to the graph knots in

Γ(K) show that K is k-mate.

Theorem 9. Suppose that f is an ordered k-function. Then, every critical knot K of f is

a graph knot. Moreover, there exist a graph kit Φ of K such that:

(i) Each graph knot P in Φ is isotopic to the core of a solid torus RP where ∂RP ⊆ f−1(r)

for some regular value r of f .

(ii) For each nontrivial graph knot P in Φ, there exists a solid torus RΓ(P ) such that the

28

core of RΓ(P ) is isotopic to

J and also, ∂RΓ(P ) ⊆ f−1(r) for some regular

value r of f . Moreover, the core of RP can be isotoped into ∂RΓ(P ) in S3.

#

J∈ΓΦ(P )

(iii) There exists a solid torus RΓ(K) such that the core of RΓ(K) is isotopic to

J
J∈ΓΦ(K)
and also, ∂RΓ(K) ⊆ f−1(r) for some regular value r of f . Moreover, K can be isotoped

#

into ∂RΓ(K).

Remark. Part (i) of Theorem 9 does not say that a saddle K of an ordered k-function f is
the core of a solid torus R where ∂R ⊆ f−1(r) for some regular value r of f but only that

there exists a graph kit φ of K such that this is true for every graph knot in Φ. However, K

is not in Φ.

Proof. There is nothing to prove if K is trivial because we can take the empty set as a graph

kit of K. If f has no saddles, then f has a single unknot source and a single unknot sink

which form a Hopf link by Lemma 3 and the theorem holds in this case. Assume now that
K is nontrivial so that f has saddles. Let a1 < ··· < aj < b1 < ··· < bk < c1 < ··· < cm

denote the critical values of f where ai, bi and ci correspond to a source, a saddle and a sink

of f respectively. We will apply the proof technique in Lemma 7 to induct on the number

k of saddles of f . As we will cover the similar cases or subcases, we will omit some details

which can be found in that proof.

Let K1 be the saddle of f with f (K1) = b1. Let A be the stable annulus of K1 in
f−1([b1− , b1]) and C1 and C2 be the stable circles of K1 in f−1(b1− ). The circles C1 and

C2 are cable knots of (not necessarily distinct) sources P1 and P2 respectively. Let Ei denote
the solid torus component of f−1([a1, b1 − ]) containing Pi. Let E denote the component
of f−1([a1, b1 + ]) which contains P1 ∪ P2 ∪ K1.
Case 1. Only one of C1 and C2 bounds a disk in f−1(b1 − ).

29

Say, C1 bounds a disk D in f−1(b1 − ) Then, K1 and P2 are unknots so that K is

distinct from K1 and P2 and also, C2 is a longitude of P2.

If P1 (cid:54)= P2, then E is isotopic to E1 in S3. We deﬁne f1 by f1(p) := f (p) for p /∈ E

so that E becomes a k-model neighborhood of the source P1 of f1. We apply the induction

hypothesis to the critical knot K of f1 to conclude that K is a graph knot and also, there

exists a graph kit Φ of K satisfying the properties stated in the theorem for f1 where these

properties include a collection of various solid tori in S3. The boundary ∂Ω of one of these
solid tori Ω is in the preimage of a regular value of f except possibly when ∂Ω ⊆ E but

then, the solid torus Ω is a tubular neighborhood of P1 and we can ﬁnd another appropriate
tubular neighborhood ˜Ω of P1 in E such that ∂ ˜Ω ⊆ f−1(aj + ). Therefore, the graph

kit Φ together with a slightly modiﬁed collection of solid tori (if necessary) works for the

k-function f as well.

If P1 = P2, then E is a solid torus not containing K. Let KE denote the core of E.
We deﬁne f1 by f1(p) := f (p) for p /∈ E so that E becomes a k-model neighborhood of the
source KE of f1. The rest of the proof continues just as in the previous P1 (cid:54)= P2 situation.
Case 2. Both C1 and C2 bound disks D1 and D2 in f−1(b1 − ) respectively.

Then, K1 is an unknot so that K (cid:54)= K1. First assume that P1 (cid:54)= P2 so that E is a

connected sum of two solid tori. Let B1 and B2 denote the 3-balls bounded by the sphere
D1 ∪ D2 ∪ A in S3 where the cut of E along this sphere produces two punctured solid tori
in S3. Say, K ⊆ Int(B1) and also, say P2 ⊆ B2. The region E ∪ B2 is then isotopic to E1
in S3. We deﬁne f1 by f1(p) := f (p) for p /∈ E ∪ B2 so that E ∪ B2 becomes a k-model

neighborhood of the source P1 of f1. We apply the induction hypothesis to the critical knot

K of f1 to prove that K is a graph knot and also, it has a graph kit Φ together with various

30

collection of solid tori satisfying the properties of the theorem for f1. This collection of solid
tori will work for f after modifying the ones in E ∪ B2 if necessary.

We now consider the situation P1 = P2. The disks D1 and D2 are not disjoint and say,
D2 ⊆ D1. Let A1 denote the annulus D1 − Int(D2). The torus T0 = A1 ∪ A separates
S3 into two closed regions and let R0 denote the one of them such that Int(R0) ∩ E1 = ∅.
Similarly, let ˜R0 denote the component of S3 − Int(E) such that ˜R0 is isotopic to R0 in S3.
The region R0 is diﬀeomorphic to the complement of a knot K0 in T0. Since E1 ∪ R0 is
isotopic to E1 in S3, we can deﬁne an ordered k-function ˜f by modifying f within a small
neighborhood U of R0 ∪ D1 such that ˜f (p) = f (p) for p /∈ U and also, ˜f does not have any

critical points in U . So, the saddle K1 of f in U is removed.

If K (cid:42) R0, we deﬁne f1 by f1(p) := ˜f (p). The induction hypothesis applies to the

critical knot K of f1 so that K is a graph knot and also, there exists a graph kit Φ of K

such that the solid tori corresponding to the graph knots in Φ satisfy the properties stated in

the theorem for f1. For the boundary of one of those solid tori lying in f−1
1 (r) for a regular
value r of f1, we will have f−1
1 (r) (cid:42) f−1(r)
for some regular value r with r ≤ b1 +  but then we can use another appropriate choice of
regular value r0 ≤ b1 +  instead of r and we can achieve f−1
1 (r0) = f−1(r0). Therefore, the

1 (r) ⊆ f−1(r) if r > b1 + . We may have f−1

graph kit Φ of K satisﬁes the properties stated in the theorem for f as well.

If K ⊆ R0, we deﬁne f1 by f1(p) := f (p) for p ∈ ˜R0 so that S3 − ˜R0 becomes a k-model

neighborhood of the source K0 of f1. We apply the induction hypothesis just as before to

conclude that K is a graph knot and also, there exists a graph kit Φ of K satisfying the

properties stated in the theorem for f1. That graph kit will satisfy those properties for f as

well except that there may be a tubular neighborhood of K0 associated to a graph knot in Φ.

This tubular neighborhood may not exactly work for f but it is isotopic to an appropriate

31

tubular neighborhood S3 − Int( ˜R0) coming from f .
Case 3. None of C1 and C2 bounds a disk in f−1(b1 − ).

Subcase 1. Both C1 and C2 bound meridian disks D1 and D2 of P1 and P2 respectively.
Then, K1 is an unknot with K (cid:54)= K1. We have P1 = P2 and the sphere S1 := D1∪D2∪A
yields P1 (cid:39) Pa#Pb. The region S3 − E has two components Ra and Rb which are isotopic
to the complement of Pa and Pb in S3 respectively.

Assume K (cid:54)= P1. Say, K ⊆ Ra. We deﬁne f1 by f1(p) := f (p) for p ∈ Ra so that
S3 − Ra becomes a k-model neighborhood of the source Pa of f1. We then apply the

induction hypothesis just as before.

Assume now K = P1. We can use f1 above and similarly deﬁne f2 for the regions Rb
and S3 − Rb to conclude that Pa and Pb are graph knots and also, there exist graph kits Φa

and Φb of Pa and Pb respectively such that Φa and Φb satisfy the properties stated in the
theorem for f1 and f2 respectively. Since P1 (cid:39) Pa#Pb, the source P1 is also a graph knot
and Φ := Φa ∪ Φb ∪ {Pa, Pb} is a graph kit of P1.

For each graph knot P in Φa ∪ Φb, we already have a solid torus RP or RΓ(P ) associated
to it. The boundaries ∂RP or ∂RΓ(P ) are then in f−1(r) for some regular value r of f

except possibly when RP or RΓ(P ) is a tubular neighborhood of Pa or Pb but this problem

in those exceptional cases can be easily resolved by just picking a more appropriate tubular

neighborhood RP or RΓ(P ) of Pa or Pb in the beginning. For the graph knots Pa and Pb
in Φ, we associate the solid tori E ∪ Rb and E ∪ Ra to RPa and RPb
we regard K as the cable knot (K)1,q of itself and deﬁne RΓ(K) := E1. The collection of

respectively. Finally,

all these solid tori associated to the graph knots in the graph kit Φ of K satisﬁes then the

properties stated in the theorem for f .

32

Subcase 2. Only C1 bounds a meridian disk D1 of P1.

Then, K1 is an unknot so that K (cid:54)= K1. The sources P1 and P2 are distinct since C2 is

a non-meridian, nontrivial cable knot of P2. Also, P2 is an unknot and C2 is a longitude of

P2. The region E is isotopic to E1 in S3. This subcase is then similar to Case 1 and the

proof in this case can be completed similarly.

Subcase 3. None of C1 and C2 is a meridian of P1 and P2 respectively.

We will ﬁrst consider the situation P1 (cid:54)= P2. The isotopic cable knots C1 and C2 are,

say, C1 (cid:39) (P1)p,q and C2 (cid:39) (P2)r,s where p, r (cid:54)= 0 as Ci is not a meridian of Pi.

If p or r is equal to 1, then P1 (cid:31) P2 or P2 (cid:31) P1 and E is a tubular neighborhood of P1
or P2. Say, Pa (cid:31) Pb where {Pa, Pb} = {P1, P2}. Let {Ea, Eb} := {E1, E2} be such that Ea

and Eb are tubular neighborhoods of Pa and Pb respectively. We deﬁne f1 by f1(p) := f (p)
for p ∈ (S3 − E) ∪ Eb so that E becomes a k-model neighborhood of the source Pb of f1. If

K is distinct from K1 and Pa, we can then apply the induction hypothesis to critical knot K
of f1 to prove the theorem. If K is equal to Pa or K1, then K (cid:31) Pb. We apply the induction

hypothesis to the source Pb of f1 to conclude that Pb is a graph knot and there exists a
graph kit Φb of Pb satisfying the properties stated in the theorem for f1. Since K (cid:31) Pb, the
knot K is also a graph knot and also, Φb ∪ {Pb} is a graph kit of K. The solid tori RP or

RΓ(P ) is already deﬁned for a graph knot P in Φb. We deﬁne RPb

:= Eb and RΓ(K) := Eb.

The collection of these solid tori satisﬁes then the properties stated in the theorem for f .

If p, r (cid:54)= 0,±1, then P1 ∪ P2 is a Hopf link so that nontrivial K is distinct from P1 and
P2. If K = K1, the saddle K is a torus knot. The graph kit {P1} of K1 together with the

solid tori RP1
:= E1 and RΓ(K) := E1 proves the theorem. Assume now that K is inside the
solid torus V := S3 − Int(E) where the core KV of V is a nontrivial torus knot (cid:39) (P1)p,q.

33

This is the situation where we will need the fact from Lemma 7 and also, the utility of

a graph kit satisfying the properties stated in the theorem as we now embed V into S3 by
g : V → S3 such that g(V ) becomes a standard, unknotted solid torus in S3. We deﬁne f1
by f1(p) := f (g−1(p)) for p ∈ g(V ) so that S3 − g(V ) becomes a k-model neighborhood of

an unknot source Kg of f1.

If g(K) is trivial, then Kg and g(K) are cable knots of each other by Lemma 7. Therefore,
K is a nontrivial, non-meridian cable knot of KV . The graph kit {KV , P1} of K together

with the solid tori RKV

:= V, RΓ(KV ) := E1, RP1

:= E1 and RΓ(K) := V proves the

theorem.

Assume now that g(K) is nontrivial. An application of the induction hypothesis to the

critical knot g(K) of f1 shows that g(K) is a graph knot and also, it produces a graph kit

Φg of g(K) satisfying the properties stated in the theorem for f1. For P in Φg, let RP and

RΓ(P ) (when P is nontrivial) be the solid tori as stated in the theorem. Let RΓ(g(K)) be the

solid torus for ΓΦg (g(K)) as stated in the theorem. If RP is not a tubular neighborhood of
Kg, we can assume RP ⊆ g(V ). If RP is a tubular neighborhood of Kg, then the standard

solid torus RP can be replaced by the standard solid torus g(V ) because the unknotted solid

tori RP and g(V ) are isotopic in S3 and also, a cable knot of the unknot core of RP is a
cable knot of the unknot core of g(V ) as well. Hence, we can assume that RP ⊆ g(V ) for
each P ∈ Φg. Similarly, we can assume that RΓ(P ) ⊆ g(V ) for each nontrivial P ∈ Φg and
also, RΓ(g(K)) ⊆ g(V ).

Let KP denote the core of RP for P ∈ Φg. Since the graph kit Φg is a collection of
isotopy classes of knots, the knot g−1(P ) is not deﬁned. However, the knot g−1(KP ) is well

deﬁned and it will do the job. If KP is an unknot, then Lemma 7 asserts that KP is a cable
knot of Kg. Therefore, the knot g−1(KP ), which is the core of the solid torus g−1(RP ), is a

34

cable knot of KV so that it is a graph knot. Let Φ1 := {g−1(KP ) : P ∈ Φg}. The properties
of Φg stated in the theorem imply that every knot in Φ1 is a graph knot because g−1(KP )

is a graph knot for every unknot P in Φg.

Let Φ0 := {P ∈ Φg : KP is trivial but g−1(KP ) is not trivial}. Let Φ2 := Φ1∪{(KV )P :
P ∈ Φ0} ∪ {(P1)P : P ∈ Φ0}, where the latter two sets contain just distinct copies of the
same knots KV and P1. For each nontrivial graph knot g−1(KP ) in Φ1, we take Γ(g−1(KP ))
as {g−1(KJ ) : J ∈ ΓΦg (P )} (when P is nontrivial) or {(KV )P} (when P is trivial). We
also take Γ((KV )P ) as {(P1)P}. So, we have Γ(J) ⊆ Φ2 when J is a nontrivial graph knot
in Φ2. To each graph knot g−1(KP ) in Φ1, we associate the solid torus g−1(RP ). To each

graph knot (KV )P or (P1)P in Φ2, we associate the solid tori V or E1 respectively. When
g−1(KP ) in Φ2 is nontrivial, we take the solid torus g−1(RΓ(P )) (when P is nontrivial) or V
(when P is trivial) for RΓ(g−1(KP )). Finally, we deﬁne RΓ((KV )P ) := E1 for (KV )P ∈ Φ2
and RΓ(K) := g−1(RΓ(g(K))). The collection of these solid tori associated to the trivial or

nontrivial graph knots in Φ2 satisﬁes the properties stated in the theorem for the ordered
k-function f . These properties of the solid tori imply now that the knot g−1(g(K)) = K is

a graph knot that is woven from the graph knots in Φ2.

We will now prove the theorem for the situation P1 = P2. The nontrivial circles C1

and C2 separates ∂E1 into two closed annuli A1 and A2 and the components of s(∂E1) are
isotopic to the tori Σ1 := A1 ∪ A and Σ2 := A2 ∪ A in S3. Let Hi denote the closed region
bounded by Σi in S3 such that Int(Hi) ∩ E1 = ∅. Similarly, let ˜Hi denote the component
of S3 − Int(E) that is isotopic to Hi in S3. Then, at least one of H1 and H2, say H1, is a
solid torus. Let KH denote the core of H1. The link P1 ∪ KH is either a Hopf link with C1

being a nontrivial torus knot or one of P1 and KH is a cable knot of the other one. In the
latter cable knot cases, the region H1 ∪ E1 is isotopic to H1 or E1 in S3.

35

There are now several possibilities for the location of K in regard of ˜H1∪E ⊆ S3. We start
with the assumption K (cid:42) ˜H1 ∪ E. If P1 (cid:31) KH or KH (cid:31) P1, let {Ka, Kb} := {P1, KH}
be such that Ka (cid:31) Kb. We deﬁne f1 by f1(p) := f (p) for p /∈ ˜H1 ∪ E so that ˜H1 ∪ E

becomes a k-model neighborhood of the source Kb of f1. An application of the induction
If P1 ∪ KH is a Hopf link, the region S3 − ˜H1 ∪ E is a

hypothesis proves the theorem.

tubular neighborhood of a nontrivial torus knot isotopic to K1 and we have already proven
the theorem in this setting which was analyzed in the situation P1 (cid:54)= P2.

Assume now K ⊆ E so that K is equal to P1 or K1. We ﬁrst consider K = P1. As
K is nontrivial, P1 ∪ KH cannot be a Hopf link so that H1 ∪ E1 is isotopic to H1 or E1
in S3. Let {Ka, Kb} and f1 be deﬁned just as in the previous paragraph. The induction

hypothesis applies to the source Kb of f1 so that Kb is a graph knot and also, there exists a

graph kit Φb of Kb satisfying the properties stated in the theorem for f1. If K = Kb, then
this graph kit Φb works for f as well. Otherwise, K (cid:31) Kb so that K is a graph knot and
Φ := Φb ∪ {Kb} is a graph kit of K. Set the solid tori RKb
:= ˜H1 and RΓ(K) := ˜H1. The

collection of these two solid tori together with the solid tori associated to the graph knots

in Φb proves the theorem.

Assume now K = K1. Our previous argument shows that P1 is a graph knot and when

P1 is not trivial, there exists a graph kit Φ1 of P1 satisfying the properties stated in the
theorem. If P1 is trivial, then take Φ1 to be the empty set. Since K (cid:31) P1, the knot K is a
graph knot and Φ1 ∪{P1} is a graph kit of K. The collection of the solid tori corresponding

to the graph knots in Φ1 together with RP1

:= E1 and RΓ(K) := E1 proves the theorem.

Assume now K ⊆ ˜H1.

If P1 ∪ KH is a Hopf link, we deﬁne f1 by f1(p) := f (p) for
p ∈ ˜H1 so that S3 − ˜H1 becomes a k-model neighborhood of the source P1. We then apply
the induction hypothesis as usual. If P1 (cid:31) KH or KH (cid:31) P1, we can deﬁne {Ka, Kb} and f1

36

just as before and our previous argument shows that Kb is a graph knot. Also, we get a graph
kit Φb of Kb satisfying the properties stated in the theorem for f1. Let {Ua, Ub} := {E1, ˜H1}

be such that Ua and Ub are tubular neighborhoods of Ka and Kb respectively. If KH = Kb,
we simply deﬁne ΦH := Φb. If KH = Ka, we deﬁne a graph kit ΦH := Φb∪{Kb} of KH and

the solid tori RKb
:= Ub and RΓ(KH ) := Ub. The graph kit ΦH of KH with its associated
solid tori satisﬁes then the properties stated in the theorem. We now embed ˜H1 into S3 by
g : ˜H1 → S3 such that g( ˜H1) becomes a standard, unknotted solid torus in S3. Such an
embedding g onto a standard, unknotted solid torus has been studied before in the P1 (cid:54)= P2

situation. We can similarly prove the theorem in this situation by combining the graph kits

ΦH and Φg of g(K). This completes the proof of the theorem.

Proof of Theorem 1. Lemma 8 proves one side of the theorem and Theorem 9 proves the

other side since any k-function f can be made ordered by modifying it within a tubular

neighborhood of its critical link without changing the set of critical points of f .

37

Chapter 5

Conclusion

The classiﬁcations of graph knots in [11], [12] are better than ours in several respects. Their

classiﬁcations are stronger as they classify all the graph links in S3 or in a homology 3-sphere.

Such a classiﬁcation attempt demands a global picture of the whole graph link rather than

a small picture of a component of a graph link. Our own narrow perspective to classify just

the graph knots but not the graph links has limited us to work in a smaller, local setting

with deﬁcient information where we have gathered extra technical machinery (Lemma 7 and

parts (i), (ii) and (iii) of Theorem 9) to overcome these deﬁciencies. As such deﬁciencies do

not exist in the global settings in [11] and [12], their studies and proofs seem more natural

than ours.

We conclude our work with the following ﬁnal remarks. If K and J are two non-isotopic

graph knots, then one way to qualitatively distinguish them is to look at the minimal numbers
k and j such that K ∈ Sk and J ∈ Sj. The bigger the diﬀerence |k − j| gets, then one can

interpret that K and J become more distinct from each other.

Suppose now that a knot K is not a graph knot so that we don’t have a k-function

to study it directly. How can we measure its deviation from being a graph knot? One

trivial approach is to consider all the knot diagrams of K. The over or under crossings of

a given diagram can be interchanged until the produced diagram becomes an unknot where

an unknot is a graph knot. Therefore, there exists a minimal positive integer n (similar to

38

the unknotting number of K) such that one produces a graph knot after making n crossing

changes on a knot diagram of K. The bigger n gets, then one can think that K deviates

more from being a graph knot.

The perspective of [11] gives a conceptually better answer to our above question. When

K is not a graph knot, the JSJ-decomposition of the complement of K has at least one

atoroidal (non-Seifert ﬁbered) piece. The more atoroidal pieces there are, the more K devi-

ates from being a graph knot.

39

REFERENCES

40

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42