Minimal coloring numbers of

Z-colorable links

Division of Mathematical Sciences,

Department of Earth Information Mathematical Sciences,

Graduate School of Integrated Basic Sciences, Nihon University

6116D01

Eri Matsudo

January, 2019 2 Contents

1 Introduction 5

2 Preliminaries 11

2.1 , links and their diagrams ...... 11

2.2 Colorings ...... 15

2.3 Coloring matrix and Determinants ...... 17

2.4 Minimal coloring number ...... 22

2.5 Some properties of Z-colorings ...... 24

3 Simple Z-coloring 31

3.1 Simple Z-coloring ...... 31

3 4 CONTENTS

3.2 Links with simple Z-colorings ...... 39

4 Minimal coloring number is four 47

5 Parallels 59

6 Minimal diagrams 77

6.1 Pretzel links ...... 78

6.2 Torus links ...... 86 Chapter 1

Introduction

In [3], Fox introduced one of the most well-known invariants for knots and links in the 3-sphere, which is now called the Fox n-coloring, or simply n- coloring for n ≥ 2. See Definition 2.7. For example, the is much often used to prove that the trefoil is a non-trivial . However, some of links are known to admit non-trivial n-colorings for an integer any n ≥ 2. In fact, if the determinant of a L is 0, then it is shown that L admits a non-trivial n-coloring for any n ≥ 2. See Section 2.3. In that case, L admits a generalization of a Fox n-coloring, which we call a

5 6 CHAPTER 1. INTRODUCTION

Z-coloring.

Definition 1.1. Let L be a link and D a regular diagram of L. We consider a map γ : {arc of D} → Z. If γ satisfies the condition 2γ(a) =

γ(b) + γ(c) at each crossing of D with the over arc a and the under arcs b and c, then γ is called a Z-coloring on D.A Z-coloring which assigns the same value to all the arcs of the diagram is called a trivial Z-coloring.

A link is called Z-colorable if it has a diagram admitting a non-trivial

Z-coloring.

Throughout the following, we call the integers of the image of a Z- coloring colors.

There are a lot of researches on Fox colorings of links. Among them, in

[5], Harary and Kauffman originally defined the minimal coloring number for a link as the minimal number of colors used in Fox colorings of the link.

Since then, it has been studied in details by many researchers. Actually, the minimal coloring numbers for knots and links behave interestingly, and are often hard to determine. 7

Here we define the minimal coloring number for Z-colorable links as a generalization.

Definition 1.2. Let us consider the cardinality of the image of a non- trivial Z-coloring on a diagram of a Z-colorable link L. We call the minimum of such cardinalities among all non-trivial Z-colorings on all diagrams of L the minimal coloring number of L, and denote it by

mincolZ(L).

In Chapter 2, we collect some preliminary results on Z-colorings. For example, we see that the minimal coloring number of any splittable Z- colorable link is 2, and for a non-splittable Z-colorable link L, we show

that mincolZ(L) ≥ 4. That is, there are no Z-colorable links with the minimal coloring number 3. Next, based on observations for Z-colorable link with at most 9 crossings, we introduce a simple Z-coloring in Section

3.1, and show that if a non-splittable link L admits a simple Z-coloring,

then mincolZ(L) = 4. This chapter is based on [7].

In Chapter 3 and 4, we actually show that any non-splittable Z- 8 CHAPTER 1. INTRODUCTION colorable link has a diagram with a simple Z-coloring, and its minimal coloring number is always four. The result is given in [?]. Remark that this result is also proved by Meiqiao Zhang, Xian’an Jin and Qingying

Deng almost independently in [20].

In the proof of the above result, we give a procedure to obtain a diagram with a Z-coloring of four colors from any given diagram with a non-trivial Z-coloring of a non-splittable Z-colorable link. However, from a given diagram of a Z-colorable link, by using the procedure given in our proof, the obtained diagram and Z-coloring would be very complicated.

In Chapter 5, we give “simple” diagrams with Z-colorings of four colors for some particular class of Z-colorable link. In fact, we consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link. This chapter is based on in [15].

Finally, in Chapter 6, we consider the question, for a Z-colorable link, when “simple” diagrams admit Z-colorings with only 4 colors. Actually, 9 we consider the minimal coloring numbers of minimal diagrams of Z- colorable links, that is, the diagrams representing the link with least number of crossings. We first show that, for any positive integer N, there exists a non-splittable Z-colorable link with a minimal diagram admitting only Z-colorings with at least N colors. In fact, the examples are given by families of pretzel links; P (n, −n, n, −n, . . . , n, −n) with at least 4 strands, P (−n, n + 1, n(n + 1)) with a positive integer n. On the other hand, by considering some particular subfamily, as a corollary, we have the following. There exists an infinite family of Z-colorable pretzel links each of which has a minimal diagram admitting a Z-coloring with only four colors. Also we give such examples given by some of Z-colorable torus links. This chapter is based on in [8] and containts some ongoing works. 10 CHAPTER 1. INTRODUCTION Chapter 2

Preliminaries

In this chapter, after recalling fundamentals in , we introduce

Z-colorings of links and state their basic properties.

2.1 Knots, links and their diagrams

In this section, we recall fundamentals in knot theory.

Definition 2.1. Let S1,...,Sm be the copies of the 1-dimensional spheres

for a natural number m. For S = S1 ∪ S2 ∪ · · · ∪ Sm and an embedding

11 12 CHAPTER 2. PRELIMINARIES f : S → R3, we define an m-component link as f(S). In particular, a

1-component link is called a knot.

Definition 2.2. Consider a link L = f(S) and a projection π : R3 → R3

3 defined by (x1, x2, x3) 7→ (x1, x2, 0). For a composition π ◦ f : S → R , if

π ◦ f(S) has only transverse double points, then π ◦ f is called a regular projection map of L and Lˆ = π(L) is called a regular projection of L. We introduce crossing information to Lˆ from L as illustrated in Figure 2.1.

A picture obtained in this way is called a regular diagram (or a diagram) of L.

Figure 2.1: crossing information

Definition 2.3. If there exists a homeomorphism H : R3 × I → R3 × I such that H(x, 0) = (x, 0) and H(L, 1) = (L0, 1), then we say that L and 2.1. KNOTS, LINKS AND THEIR DIAGRAMS 13

L0 are ambient isotopic.

There exist some definitions of equivalence of links. See [11], for example.

Definition 2.4. The transformations shown in Figure 2.2, Figure 2.3 and

Figure 2.4 are called the . There we are supposing that the diagrams on the outsides of the circles are identical before and after the moves.

Figure 2.2: The Reidemeister move of type 1

Figure 2.3: The Reidemeister move of type 2 14 CHAPTER 2. PRELIMINARIES

Figure 2.4: The Reidemeister move of type 3

It is not easy to show whether two links are ambient isotopic or not in R3. In 1927, Reidemeister proved the next theorem. See [11], for example.

Theorem 2.5. A diagram of a link can be transformed to a diagram of another link by a finite sequence of Reidemeister moves if and only if the links represented by the diagrams are ambient isotopic.

Definition 2.6. A link admitting a 2-sphere embedded in the exterior which separates one or more components of the link from the others is 2.2. COLORINGS 15 called a splittable link. A link which is not splittable is called a non- splittable link.

2.2 Colorings

In 1961, Fox introduced an invariant for knots and links in [3]. The invariant is called the Fox n-coloring or simply an n-coloring for a natural number n, and it is one of the most well-known invariants for knots and links.

Definition 2.7. Let L be a link and D a regular diagram of L. We consider a map γ : {arcs of D} → Z. For a natural number n, if γ satisfies the condition 2γ(a) ≡ γ(b) + γ(c) (mod n) at each crossing of

D with the over arc a and the under arcs b and c, then γ is called an n- coloring on D. An n-coloring which assigns the same color to all the arcs of the diagram is called the trivial n-coloring. A link is called n-colorable for a natural number n if it has a diagram admitting a non-trivial n- 16 CHAPTER 2. PRELIMINARIES coloring.

A Z-coloring is defined as a generalization of a Fox n-coloring as given

Definition 1.1.

The links illustrated in Figure 2.5 is an example which is Z-colorable.

Throughout this paper, we adopt the names of links as those given in [2].

Figure 2.5: L8n6 (a ≥ 1)

In the figure, the number 0, a, 2a, 4a and 4a denote the color assigned to arc near the number by a Z-coloring. 2.3. COLORING MATRIX AND DETERMINANTS 17 2.3 Coloring matrix and Determinants

In this section, we recall the definition of the determinant det L of a link

L, and give a proof of the fact that a link L is Z-colorable if and only if det L is 0.

Let us consider a system of homogeneous linear equations modulo n by regarding the arcs of a diagram D of a link L as algebraic variables and by setting up the equation at each crossing as: twice the over arc minus the sum of the under arcs equals zero. This system of equations is called the coloring system of equations for D. Then there is a natural correspondence between Z-colorings of D and solutions of the coloring system of equations for D.

We obtain the coefficient matrix of the coloring system of equations for a Z-coloring γ on D. Precisely, associated to γ, we obtain a matrix

A = (aij) with aij ∈ Z, which is a n × n matrix, where n denotes the 18 CHAPTER 2. PRELIMINARIES number of the crossing on D as follows.

   −2 if the j-th arc is the over arc on the i-th crossing,     aij = 1 if the j-th arc is the under arc on the i-th crossing, and      0 otherwise. 

Then the matrix A∗ obtained by deleting one row and one column from A is called the coloring matrix. It is known that the absolute value of the coloring matrix for D gives an invariant of the link L, which is called the determinant of L, denoted by det L. See [11] for example.

Now we can see that there are non-trivial Z-coloring of a link L if and only if det L = 0.

Theorem 2.8. A link L is Z-colorable if and only if det L is 0.

Proof. First, we assume that det L is 0. Since a link L has a diagram 2.3. COLORING MATRIX AND DETERMINANTS 19 admitting a trivial Z-coloring, we see

   1         .  A  .  = O.        1 

It follows that the row vectors of A are linearly dependent. That is, there

exist c1, . . . , cn ∈ Z such that c1(a1,1 . . . a1,n) + c2(a2,1 . . . a2,n) + ··· + cn(an,1 . . . an,n) = O and

 

 c1         .   .  6= O.         cn

From there exists cj 6= 0, we can exchange the labels to j = 1. Then

we see that (a1,1 . . . a1,n) = −c2/c1(a2,1 . . . a2,n) − · · · − cn/c1(an,1 . . . an,n).

Let A∗ be a coloring matrix by deleting the 1st row and the 1st column from A. From the assumption, det A∗ is 0. There exists a vector x = 20 CHAPTER 2. PRELIMINARIES t ∗ (x2 . . . xn) 6= O such that A x = O. That is, we see the following.

    x2a2,2 + ··· + xna2,n = 0    . .      x2an,2 + ··· + xnan,n = 0

0 t 0 Then for x = (0 x2 . . . xn), we see that (ak,1 . . . ak,n)x = 0 for any

2 ≤ k ≤ n, and have the following.

x2a1,2 + ··· + xna1,n = (a1,2 . . . a1,n)x

0 = (a1,1 a1,2 . . . a1,n)x

0 = (−c2/c1(a2,1 . . . a2,n) − · · · − cn/c1(an,1 . . . an,n))x

0 0 = −c2/c1(a2,1 . . . a2,n)x − · · · − cn/c1(an,1 . . . an,n)x

= 0

Therefore Ax0 = O and x 6= O, x 6= t(α . . . α) for any α ∈ Z.

0 t Next, we assume L is Z-colorable. There exists x = (x1 x2 . . . xn) 2.3. COLORING MATRIX AND DETERMINANTS 21 such that Ax0 = O and x 6= O, x 6= t(α . . . α) for any α ∈ Z. Let x be

t ∗ the vector (x2 − x1 . . . xn − x1) and A a coloring matrix by deleting the

1st row and the 1st column from A. Here we see that

0 t O = A(x − (x1 . . . x1))

t = A (0 x2 − x1 . . . xn − x1)  

 0 + (x2 − x1)a1,2 + ··· + (xn − x1)a1,n           0 + (x2 − x1)a2,2 + ··· + (xn − x1)a2,n    =      .   .          0 + (x2 − x1)an,2 + ··· + (xn − x1)an,n

Therefore, we see the following.

 

 (x2 − x1)a2,2 + ··· + (xn − x1)a2,n        ∗  .  A x =  .          (x2 − x1)an,2 + ··· + (xn − x1)an,n

= O 22 CHAPTER 2. PRELIMINARIES

We obtain det A∗ is 0 from x 6= 0, that is, det L is 0.

2.4 Minimal coloring number

In [5], Harary and Kauffman defined the minimal coloring number for links with Fox n-colorings as follows.

Definition 2.9. Let us consider the cardinality of the image of a non- trivial Fox n-coloring on a diagram of a Fox n-colorable link L. We call the minimum of such cardinalities among all non-trivial Fox n-colorings on all diagrams of L the minimal coloring number of L, and denote it by

mincoln(L).

Furthermore, in [19], Satoh found that any 5-colorable knot has a non-trivially 5-colored diagram with exactly four colors. In [18], Oshiro found that any 7-colorable knot has a non-trivially 7-colored diagram with exactly four colors. In [17], Nakamura, Nakanishi and Satoh found 2.4. MINIMAL COLORING NUMBER 23 that any 9-colorable knot has a non-trivially effective 9-colored diagram with exactly five colors.

In general, for any Fox n-colorable link, we found the following result in [6].

Theorem 2.10. Let n be a natural number. For any n-colorable link

L with non-zero determinant, let mincoln(L) be the minimal number of

colors on effectively n-colored diagrams of L. Then mincoln(L) ≥ 1 +

log2 n holds.

Here we define the minimal coloring number for Z-colorable links as a generalization.

Definition 2.11. Let us consider the cardinality of the image of a non- trivial Z-coloring on a diagram of a Z-colorable link L. We call the minimum of such cardinalities among all non-trivial Z-colorings on all diagrams of L the minimal coloring number of L, and denote it by

mincolZ(L). 24 CHAPTER 2. PRELIMINARIES

Throughout this thesis, when a Z-coloring is given on a diagram of a link, we say that a crossing is colored by {a|b|c} if the over arc colored by b and the under arcs colored by a and c.

2.5 Some properties of Z-colorings

For any Z-colorable link, we have the following about mincolZ(L) shown in [7].

Lemma 2.12. For any Z-colorable link, there exists a Z-coloring γ such

that Im(γ) = {0, a1, a2, . . . , an} with ai > 0 (i = 1, 2, . . . , n) for some positive integer n.

Proof. We assume that there exists a Z-coloring γ0 such that Im(γ0) =

{α, b1, b2, . . . , bn} with bi > α (i = 1, 2, . . . , n) for some positive integer n. We consider a map γ : {arc of the diagram} → Z with Im(γ) =

{0, a1, a2, . . . , an} obtained by setting ai = bi − α. Then at a crossing on the diagram of the link, the colors {p|q|r} are transformed to {p − 2.5. SOME PROPERTIES OF Z-COLORINGS 25

α|q − α|r − α}. We see (p − α) + (r − α) = 2(q − α) from p + r = 2q.

Therefore the map γ is a Z-coloring such that Im(γ) = {0, a1, a2, . . . , an}

with ai > 0 (i = 1, 2, . . . , n).

Lemma 2.13. For a Z-coloring γ with 0 = minIm(γ), if an over arc at a crossing is colored by 0, then the both under arcs at the crossing are colored by 0.

Proof. Suppose that a crossing has colors {a|b|c} with a 6= b 6= c for a

Z-coloring γ with 0 = minIm(γ). Then we obtain a > b > c or c > b > a.

If we suppose that b = 0, then it gives 0 > c or 0 > a, contradicting the assumption that 0 = minIm(γ).

Lemma 2.14. For a Z-coloring γ with M = max Im(γ), if an over arc at a crossing is colored by M, then the both under arcs at the crossing are colored by M.

Proof. Suppose that a crossing has colors {a|b|c} with a 6= b 6= c for a Z- coloring γ with M = maxIm(γ). Then we obtain a > b > c or c > b > a. 26 CHAPTER 2. PRELIMINARIES

If we suppose that b = M, then it gives M < c or M < a, contradicting the fact that M = maxIm(γ).

Proposition 2.15. It is seen that any splittable link L is Z-colorable and

mincolZ(L) = 2.

Proof. Let L be a splittable link. L has a diagram D = D1 ∪ D2 such

that D1 and D2 are disjoint. Here we assign the color 0 on D1, and the color 1 on D2. Then D admits a Z-coloring with only two colors 0 and

1. That is, a splittable link L has a Z-coloring with only 2 colors.

On the other hand, by using the above lemmas, we see that the next holds for non-splittable links.

Theorem 2.16. Let L be a non-splittable Z-colorable link. Then mincolZ(L) ≥

4.

Proof. Let γ be a non-trivial Z-coloring for a Z-colorable link L. We

will show that if mincolZ(L) ≤ 3, then L is splittable. Since γ is non- trivial, the cardinality of Im(γ) is greater than 1. In the case that the 2.5. SOME PROPERTIES OF Z-COLORINGS 27 cardinality of Im(γ) is 2, we can assume that Im(γ) = {0, a} by Lemma

2.12. By Lemma 2.14 a diagram of L has no crossings with over arcs labeled by a other than {a|a|a}. And it also has no crossings with over arcs labeled by 0 other than {0|0|0}. Therefore the diagram must be splittable, and we see that the link is splittable. In the case that the cardinality of Im(γ) is 3, we can assume Im(γ) = {0, a, b} by Lemma

2.12. Let b be the maximum of Im(γ). By Lemma 2.14, a diagram of the link has no crossings with over arcs labeled by b other than {b|b|b}. If the diagram has no crossing colored by {0|a|b}, it has only crossings colored by {0|0|0}, {a|a|a} or {b|b|b}. Then we obtain the link is splittable. If the diagram has a crossing colored by {0|a|b}, we see b = 2a. From

Lemma 2.13 and 2.14, the diagram has no crossings labeled by {a| ∗ |∗} with the exception of crossings labeled by {a|a|a}. Then we see that the link is splittable.

If a link is Z-colorable with four colors, we can show the following.

Theorem 2.17. If mincolZ(L) = 4 for a Z-colorable link L, then there 28 CHAPTER 2. PRELIMINARIES exists a diagram D of L and a Z-coloring γ on D such that Im(γ) =

{0, 1, 2, 3}.

Proof. Suppose that there exists a Z-coloring γ0 with four colors. From

Lemma 2.12, we can suppose Im(γ0) = {0, a, b, c} with a > 0, b > 0 and c > 0. We assume that c is the maximum of Im(γ0). Since L is a non- splittable link, we assume 0 < a < b < c. From Lemma 2.13 and Lemma

2.14, D has a crossing colored by {0|a|b} or a crossing colored by {0|b|c}.

If D has both of the crossings, by the definition of Z-coloring, we obtain a − 0 = b − a and b − 0 = c − b. We see b = 2a and c = 2b, and then we obtain Im(γ0) = {0, a, 2a, 4a}. Then the diagram has no crossings with colors {a| ∗ |∗} other than {a|a|a}. This gives a contradiction that L is a non-splittable link.

If D has only either of the crossings colored by {0|a|b} or {0|b|c}, then

D has only three colors, contradicting Theorem 2.16. From the above,

D has either of the crossings colored by {0|a|b} or {0|b|c} and another crossing colored by {a|b|c}. If D has a crossing colored by {0|b|c} and a 2.5. SOME PROPERTIES OF Z-COLORINGS 29 crossing colored by {a|b|c}, we obtain b − 0 = c − b and b − a = c − b.

We see a = 0, and this is contradictory to the assumption. If D has a crossing colored by {0|a|b} and a crossing colored by {a|b|c}, we obtain a − 0 = b − a and b − a = c − b. We see b = 2a and c = 3a. Therefore, we obtain Im(γ0)= {0, a, 2a, 3a}. By dividing by a, we have a Z-coloring with colors {0, 1, 2, 3} as desired. 30 CHAPTER 2. PRELIMINARIES Chapter 3

Simple Z-coloring

In this chapter, we focus on the “simplest” Z-coloring found for the links, which we call a simple Z-coloring. Also included are examples of

Z-colorings for Z-colorable links with at most 9 crossings.

3.1 Simple Z-coloring

For Z-colorable links, we can find the colorings on the diagrams in [2] which are quite distinctive. See Figures 3.1, 3.2, 3.3, 3.4 and 3.5.

31 32 CHAPTER 3. SIMPLE Z-COLORING

Figure 3.1: L8n6

Figure 3.2: L8n8 3.1. SIMPLE Z-COLORING 33

Figure 3.3: L9n18

Figure 3.4: L9n19 34 CHAPTER 3. SIMPLE Z-COLORING

Figure 3.5: L9n27

Based on such examples, we introduce the following notion.

Definition 3.1. Let L be a non-trivial Z-colorable link, and γ a Z- coloring on a diagram D of L. We call γ a simple Z-coloring if there exists an integer d such that, at each crossing in D, the difference between the colors of the over arc and the under arcs is d or 0.

For example, a pretzel knot P (n, −n, n, −n, . . . , n, −n) with integer 3.1. SIMPLE Z-COLORING 35 n admits a simple Z-coloring. See Figure 3.6.

Figure 3.6: Pretzel link P (n, −n, . . . , n, −n) for n ≥ 1

Then we see that, if a non-splittable link L admits a simple Z-coloring,

then mincolZ(L) = 4. This is shown in [7, Theorem 4.2].

Theorem 3.2. Let L be a non-splittable Z-colorable link. If there exists

a simple Z-coloring on a diagram of L, then mincolZ(L) = 4.

Proof. Let γ be a simple Z-coloring on a diagram of a non-splittable 36 CHAPTER 3. SIMPLE Z-COLORING

Z-colorable link L. Then there exists a natural number d such that, at all the crossing in D, the differences between the colors of the over arcs and the under arcs are d or 0. By Lemma 2.13, we suppose that 0 is the minimum of Im(γ). Let us modify the diagram and the Z-coloring into the one with simple Z-colorings. For the maximum M of Im(γ), by

Lemma 2.14, the diagram has only crossings colored by {M|M|M} or

{M|M − d|M − 2d}. First we eliminate a crossing colored by {M|M|M} as shown in Figure 3.7.

Figure 3.7:

Next, we transform the diagram inductively near the crossings colored by {M|M − d|M − 2d} as shown in Figure 3.8. 3.1. SIMPLE Z-COLORING 37

Figure 3.8:

Then we obtain a simple Z-coloring such that the maximum of colors is less than that of the previous one.

Note that after changing the diagram and the Z-coloring, the common difference of colors at each crossing is still d or 0. That is, the obtained

Z-coloring remains simple. If the maximum of colors of the obtained

Z-coloring is at least 4d, then we perform the transformation repeat- 38 CHAPTER 3. SIMPLE Z-COLORING edly. After all, we obtain a Z-coloring with {0, d, 2d, 3d}. Therefore

mincolZ(L) is at most 4. From Lemma 2.13, we see mincolZ(L) = 4.

However, there are many diagrams of Z-colorable links without simple

Z-colorings. See Figure 3.9 for example.

Figure 3.9: Non-simple Z-coloring for L10n32 3.2. LINKS WITH SIMPLE Z-COLORINGS 39 3.2 Links with simple Z-colorings

In this section, we present diagrams of the Z-colorable links whose cross- ing numbers are 10 with simple Z-colorings. Note that we give diagrams of Z-colorable links with at most 9 crossing numbers in Section 3.1. We

first use the diagrams shown in [2]. For each link, if the diagram admits only a non-simple Z-coloring, then we modify it to admit a simple one.

Figure 3.10: L10n32 with a non-simple Z-coloring 40 CHAPTER 3. SIMPLE Z-COLORING

Figure 3.11: L10n32 with a simple Z-coloring

Figure 3.12: L10n36 3.2. LINKS WITH SIMPLE Z-COLORINGS 41

Figure 3.13: L10n36 with a simple Z-coloring

Figure 3.14: L10n56 42 CHAPTER 3. SIMPLE Z-COLORING

Figure 3.15: L10n57

Figure 3.16: L10n59 with a non-simple Z-coloring 3.2. LINKS WITH SIMPLE Z-COLORINGS 43

Figure 3.17: L10n59 with a simple Z-coloring

Figure 3.18: L10n91 44 CHAPTER 3. SIMPLE Z-COLORING

Figure 3.19: L10n93

Figure 3.20: L10n94 3.2. LINKS WITH SIMPLE Z-COLORINGS 45

Figure 3.21: L10n104

Figure 3.22: L10n107 46 CHAPTER 3. SIMPLE Z-COLORING

Figure 3.23: L10n111 Chapter 4

Minimal coloring number is four

The next is the main result in this thesis.

Theorem 4.1 ([14]). The minimal coloring number of any non-splittable

Z-colorable link is equal to 4.

This result is also proved by Meiqiao Zhang, Xian’an Jin and Qingy- ing Deng almost independently in [20]. Previously Zhang gave us her

47 48 CHAPTER 4. MINIMAL COLORING NUMBER IS FOUR manuscript for her Master thesis. In Zhang’s thesis, she calls a crossing an n-diff crossing if |b−a| and |b−c| are equal to n, where the over arc is colored by b and the under arcs are colored by a and c by a Z-coloring γ at the crossing. Then she showed that if a Z-colorable link has a diagram with a 1-diff crossing, the link has a diagram with only 0-diff crossings and 1-diff crossings. Our proof is based on her arguments.

Proof of Theorem 4.1. In this proof, we depict a crossing colored by

{a|a|a} with an integer a as shown in Figure 4.1.

Figure 4.1:

And we also depict a d-diff crossing for a positive integer d as shown 49 in Figure 4.2.

Figure 4.2:

Let L be a non-splittable Z-colorable link. If the link L admits a

simple Z-coloring, by Theorem 3.2, we see that mincolZ(L) = 4. We now consider a non-simple Z-coloring for L. Let D be a diagram of L with a non-simple Z-coloring. We suppose that D has di-diff crossings for i =

1, 2, 3,... , and consider the set {0, d1, d2, . . . , dm} with the assumption

d1 ≤ d2 ≤ · · · ≤ dm. We can find a path on D from a dm-diff crossing to a

d-diff crossing passing only (possibly no) 0-diff crossings with 0 < d < dm.

Up to symmetry, such a path can be described as one of the 6 types 1-1,

1-2, 2, 3, 4-1 and 4-2 illustrated in Figure 4.3. 50 CHAPTER 4. MINIMAL COLORING NUMBER IS FOUR

Figure 4.3:

In the following, for a path emerging at a dm-diff crossing in Figure

4.3, we will modify the diagram and the coloring to eliminate the dm-diff crossing. 51

For a path of type 1-1, we modify the diagram and the coloring as shown in Figure 4.4.

Figure 4.4: Type 1-1 52 CHAPTER 4. MINIMAL COLORING NUMBER IS FOUR

For a path of type 1-2, we modify the diagram and the coloring as shown in Figure 4.5.

Figure 4.5: Type 1-2 53

For a path of type 2, we modify the diagram and the coloring as shown in Figure 4.6.

Figure 4.6: Type 2 54 CHAPTER 4. MINIMAL COLORING NUMBER IS FOUR

For a path of type 3, we modify the diagram and the coloring as shown in Figure 4.7.

Figure 4.7: Type 3 55

For a path of type 4-1, we modify the diagram and the coloring as shown in Figure 4.8.

Figure 4.8: Type 4-1 56 CHAPTER 4. MINIMAL COLORING NUMBER IS FOUR

For a path of type 4-2, we modify the diagram and the coloring as shown in Figure 4.9.

Figure 4.9: Type 4-2 57

Here the obtained diagram has |dm − d|-diff crossings, |dm − 2d|-diff

crossings and no dm-diff crossings. From 0 < d < dm, we see that |dm −d|

and |dm −2d| are less than dm. By the induction for dm, L has a diagram with a Z-coloring which has only 0-diff crossings and α-diff crossings for some α > 0. That is L admits a simple Z-coloring. By Theorem 3.2, we

conclude that mincolZ(L) = 4.

By Theorem 4.1, any non-splittable Z-colorable link has a diagram with a Z-coloring of 4 colors. However, from a given diagram of a Z- colorable link, by using the procedure given in the our proof of Theorem

4.1, the obtained diagram and Z-coloring might be very complicated. 58 CHAPTER 4. MINIMAL COLORING NUMBER IS FOUR Chapter 5

Parallels

In this chapter, we give a simple way to obtain a diagram which attains the minimal coloring number for a particular family of Z-colorable links.

We consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link, as follows.

Definition 5.1. Let L = K1 ∪ · · · ∪ Kc be a link with c components and

D a diagram of L. For a set (n1, . . . , nc) of integers ni ≥ 1, we denote by

(n1,...,nc) D the diagram obtained by taking ni-parallel copies of the i-th

59 60 CHAPTER 5. PARALLELS

(n1,...,nc) component Ki of D on the 2-sphere for 1 ≤ i ≤ c. The link L

(n1,...,nc) represented by D is called the (n1, . . . , nc)-parallel of L. When L is a knot, that is c = 1, we will say n-parallel Ln instead of (n)-parallel

L(n).

To state our results, for a 2-parallels of a knot, we prepare the follow- ing. We always orient a 2-parallel of a knot induced from an orientation of the knot.

Definition 5.2. We say a 2-parallel of a knot is untwisted if the of the 2 components of the parallel is equal to 0.

Examples of (n1, . . . , nc)-parallels of links are shown in Figure 5.1 and

Figure 5.2.

Figure 5.1: A (3, 2)-parallel of the 61

Figure 5.2: A 2-parallel of the trefoil

We show that an even parallel of a link, that is, a parallel of a link with even number of components, is Z-colorable except for the case of

2-parallels with non-zero linking number.

Theorem 5.3. [1] For a non-trivial knot K and any diagram D of K,

D2 always represents a Z-colorable link if it is untwisted. Moreover, there

2 exists a diagram D0 of K such that D0 is locally equivalent to a minimally

Z-colorable diagram.

[2] Let L be a non-splittable c-component link and D any diagram of

(n1,...,nc) L. For even numbers n1, . . . , nc at least 4, D always represents a Z-colorable link and is locally equivalent to a minimally Z-colorable 62 CHAPTER 5. PARALLELS diagram.

Here we give the definitions used in Theorem 5.3.

Definition 5.4. Let L be a Z-colorable link. A diagram D of L is called a minimally Z-colorable diagram if there exists a Z-coloring on D which attains the minimal coloring number of L.

Definition 5.5. For diagrams D and D0 of L, D is locally equivalent

0 to D if there exist mutually disjoint open subsets U1,U2,...,Un on the

2-sphere such that D0 is obtained from D by Reidemeister moves only in

Sm i=1 Ui.

To prove Theorem 5.3 [1], we prepare the next lemma about the linking number of components of 2-parallel of a knot.

Lemma 5.6. Let D a diagram of an oriented knot K. For the oriented

2 2 2-parallel K = K1 ∪ K2 represented by D , the linking number of K1

and K2 is equal to the of D. 63

Here we recall that the writhe of D is the sum of the signs of crossings on D.

Proof. Proof of Theorem 5.6 Any crossing c on D is replaced by four crossings by taking 2-parallel copies. In the four crossings, the two cross-

ings consist of the arcs of only K1 or K2, and the other two crossings

both K1 and K2.

Figure 5.3:

Then c and the crossings constructed by the arcs of different com- ponents must have same sign. See Figure 5.3 for the case of a positive

crossing. Therefore the linking number of K1 and K2 is equal to the writhe of D.

For non-splittability of parallels of knots and links, we can also show the next. 64 CHAPTER 5. PARALLELS

Lemma 5.7. [1] Any n-parallel of a non-trivial knot is non-splittable

for n ≥ 2. [2] Any (n1, . . . , nc)-parallel of a non-splittable link is non- splittable for n ≥ 1.

Proof. [1] Let Kn be an n-parallel of a non-trivial knot K. Suppose that

Kn is splittable, that is, there exists a 2-sphere S in S3 − Kn such that

S does not bound any 3-ball in S3 − Kn. On the other hand, there exists an embedded annulus A in S3 such that Kn ⊂ A and the core of A is parallel to the components of Kn.

By isotopy of S in R3 − Kn, let S ∩ A be minimized. If S ∩ A is the empty set then it is in contradiction with the definition of S and the fact that any in S3 is irreducible.

We suppose otherwise, that is, S∩A is not the empty set. We consider a component C of S ∩ A on A. Then there are two possibilities; (1) C is trivial on A, that is , C bounds a disk on A, or (2) C is parallel to a component of Kn.

In the case (1), C can be removed by isotopy of S, since the com- 65 plement of A is homeomorphic to that of K which is irreducible. That is in contradiction with that S ∩ A minimized. In the case (2), since C bounds a disk in S, K must be trivial. This also gives a contradiction.

Therefore we conclude that Kn is non-splittable.

[2] From a non-splittable link L = K1 ∪ · · · ∪ Kc with c at least 2, we

(n1,...,nc) n1 nc obtain a parallel L = K1 ∪ · · · ∪ Kc .

We assume L(n1,...,nc) is splittable. Then there exists a 2-sphere in

3 (n1,...,nc) 3 (n1,...,nc) S − L such that S = B1 ∪ B2 and L = L1 ∪ L2 with a

link Li ⊂ Bi for i = 1, 2. Here we take a component l1 from L1 and l2

from L2.

ni nj nk In the case that l1 ⊂ Ki and l2 ⊂ Kj with i 6= j. Take lk ⊂ Kk

0 S for each k 6= i, j. Then we see L = (l1 ∪ l2) ∪ k6=i,j lk is equivalent

0 to L. Now S splits l1 and l2. Since L is ambient isotopic to L, this is contradictory with the assumption.

ni nk In the case that l1, l2 ⊂ Ki . Take lk ⊂ Kk for each k 6= i. By

l1 ⊂ B1, together with the assumption that L is non-splittable, the link 66 CHAPTER 5. PARALLELS

S l1 ∪ k6=i lk is contained in B1. On the other hand, for l2, we have l2 ∪

S k6=i lk is contained in B2. This is contradictory to each other. Therefore

L(n1,...,nc) is non-splittable.

To prove Theorem 5.3, we actually give Z-colorings with four colors on the paralleled diagrams.

Proof of Theorem 5.3. [1] Let K be a non-trivial knot in S3. First we give an orientation to the knot K. Let us take a diagram D of K.

Consider the 2-parallel K2 obtained from D. Then give the orientation

2 to the 2-parallel K = K1 ∪ K2 induced from that of K. Suppose that

2 K is untwisted, i.e., the linking number of K1 and K2 is 0. Then we see the writhe of D is 0 by Lemma 2.13. Note that the number of the arcs of D with the positive crossings in both ends is equal to the same number of arcs on D with the negative crossings in both ends as shown in Figure 5.4. We consider the parallel arcs on D2 corresponding to the arcs mentioned above. 67

Figure 5.4:

Here we add a full-twist to the parallel arcs with the sign as shown in Figure 5.5 and Figure 5.6.

Figure 5.5:

Figure 5.6:

Since the writhe of D is 0, there exist positive crossings as many as negative crossings on D. Therefore the diagram obtained by this modification represents the same link K2 as for D2. In the following, 68 CHAPTER 5. PARALLELS we will use the same notation D2 to denote the modified diagram for convenience.

We will show that this D2 admits a non-trivial Z-coloring. We take a pair of parallel arcs on D2 and set the colors (integers) a and a + d to the pair of parallel arcs on D2, that is, d is a difference of colors of two parallel arcs. We give the colors to remaining arcs around the arcs to satisfy the condition of Z-coloring. See Figure 5.7. It then follows that d stays constant after passing under adjacent two parallel arcs.

Figure 5.7:

In the same way, the successive parallel arcs can be colored with only crossings shown in Figures 5.8. 69

Figure 5.8:

We take a pair of parallel arcs corresponding to the arc on D with the negative crossings in both ends, and fix the colors of two parallel arcs 0 and 1. For the arc colored by 0 is changed to be colored by 2 after passing under another two parallel arcs. See the colors as shown in Figure 5.9. Thus, the colors (0, 1) and (2, 3) appear alternately by tracing the parallel arcs along D2. We therefore obtain y = 0, 2 and y0 = 1, 3. We see that 2y = 0 or 4, 2y − 1 = −1 or 3, 2y0 − 2 = 0 or 4 and 2y0 − 3 = −1 or 3. It follows that D2 admits a Z-coloring C such that Im(C) = {−1, 0, 1, 2, 3, 4}. Therefore K2 is Z-colorable. 70 CHAPTER 5. PARALLELS

Figure 5.9:

Here we focus on the arc colored by 4 and −1. We can get a Z-coloring without the colors 4 and −1 by changing the diagram and the coloring obtained above as shown in Figure 5.10 and Figure 5.11.

Figure 5.10: 71

Figure 5.11:

We see that K2 also admits the coloring C0 such that Im(C0) =

{0, 1, 2, 3}. By Lemma 5.7, K2 is non-splittable. By [14] and [20], the obtained diagram is a minimally Z-colorable diagram, which is locally equivalent to D2.

[2] Let D(n1,...,nc) be a diagram of L(n1,...,nc). The diagram D(n1,...,nc) has only crossings shown in Figure 5.12.

Figure 5.12: 72 CHAPTER 5. PARALLELS

We put a circle as fencing the crossings as shown in Figure 5.13.

Figure 5.13:

Note that each crossing of D(n1,...,nk) is contained in some of the regions encircled.

For any parallel family of arcs (a1, . . . , ak) outside the regions, we fix

the colors of ak/2 and ak/2+1 are 1 and the others are 0 as shown in Figure

5.14. 73

Figure 5.14:

For the arcs inside the region, we assign the colors as follows.

In the case nj = 4m + 4 for some integer m, we assign the colors

−1, 0, 1, and 3 as shown in Figure 5.15.

Figure 5.15: 74 CHAPTER 5. PARALLELS

Then, at each crossing inside the region, the obtained coloring satis-

fies the condition of Z-coloring. Here the colors of the arcs intersecting the circle are compatible to those of the arcs outside.

In the case nj = 4m + 2, in the same way, we assign the colors

−1, 0, 1, 2 as shown in Figure 5.16.

Figure 5.16:

Again, at each crossing inside the region, the obtained coloring satisfy the condition of Z-coloring. Here the colors of the arcs intersecting the 75 circle are compatible to those of the arcs outside.

We see that D(n1,...,nc) admits a Z-coloring C such that Im(C) =

{−1, 0, 1, 2} or {−1, 0, 1, 2, 3}. Therefore L(n1,...,nc) is Z-colorable.

Moreover, in the case that the color 3 appears, we can delete the color

3 by changing the diagram and the coloring as shown in Figure 5.17.

Figure 5.17: Delete the color 3

Therefore there exists the coloring C0 such that Im(C0) = {−1, 0, 1, 2}.

Since we are assuming that L is non-splittable, Lemma 5.7 assures that the parallel is non-splittable. Therefore the obtained diagram is a minimally Z-colorable diagram, which is locally equivalent to D(n1,...,nc). 76 CHAPTER 5. PARALLELS Chapter 6

Minimal diagrams

In this chapter, we consider the minimal coloring numbers of minimal diagrams of Z-colorable links, that is, the diagrams representing the link with least number of crossings.

Definition 6.1. Let L be a Z-colorable link. We call the minimum of such cardinalities among all non-trivial Z-colorings on a diagram D of L

the minimal coloring number of D, and denote it by mincolZ(D).

77 78 CHAPTER 6. MINIMAL DIAGRAMS 6.1 Pretzel links

In this section, we consider Z-colorable pretzel links. Here a pretzel link

P (a1, . . . , an) is defined as a link admitting a diagram consisting rational

tangles corresponding to 1/a1, 1/a2,..., 1/an located in line. We first prove the next theorem.

Theorem 6.2. For an even integer n ≥ 2, the pretzel link P (n, −n, . . . , n, −n) with at least 4 strands has a minimal diagram admitting only Z-colorings with n + 2 colors.

Proof. The pretzel link P (n, −n, . . . , n, −n) is Z-colorable since its de- terminant is 0 for the link. See [1] for example. And since such a pretzel link P (n, −n, . . . , n, −n) is known to be non-splittable if n ≥ 2 and the number of strands is at least 4.

Let D be the diagram of P (n, −n, . . . , n, −n) illustrated in Figure

6.1 and set the labels x1, x2,... of the arcs of D. This diagram D is a minimal diagram of the link due to the result in [12]. Remark that some 6.1. PRETZEL LINKS 79 of the arcs are labelled in duplicate.

Figure 6.1:

Suppose that a non-trivial Z-coloring γ is given on D. As shown in

[7], without changing the number of the colors, we may assume that the minimum of the colors for γ is 0, and the arcs colored by 0 cannot cross over the arcs colored by the other colors. Thus, due to symmetry, we can

set the color γ(x1) of the arc x1 as 0. Also the color γ(x2) of the arc x2 80 CHAPTER 6. MINIMAL DIAGRAMS is set as a > 0, since γ is assumed to be non-trivial.

For the arcs x1, x2, . . . , xn, xn+1, xn+2 in D, we see that γ(xi) = (i−1)a

for 1 ≤ i ≤ n + 2. In particular, γ(xn+1) = na and γ(xn+2) = (n + 1)a.

0 0 0 0 0 0 Also, for x1(= x1), x2, . . . , xn, xn+1(= xn+1), xn+2, since γ(x1) = γ(x1) =

0 0 0 and γ(xn+1) = γ(xn+1) = na, we see that γ(xj) = (j − 1)a for

0 1 ≤ j ≤ n + 2. In particular, γ(xn+1) = γ(xn+1) = na.

In the same way, we can determine all the colors of the arcs on D under γ. It then follows that D is colored by γ as illustrated by Figure

6.2. 6.1. PRETZEL LINKS 81

Figure 6.2:

That is, we see mincolZ(D) = n + 2.

On the other hand, by considering some particular subfamily, as a corollary, we see that there exists an infinite family of Z-colorable pretzel links each of which has a minimal diagram admitting a Z-coloring with only four colors as follows. 82 CHAPTER 6. MINIMAL DIAGRAMS

We consider the pretzel link P (2, −2, 2, −2,..., 2, −2). The diagram depicted in Figure 6.3 is a minimal diagram by [12]. On the other hand, the Z-coloring given in the figure has only four colors {0, 1, 2, 3}.

Figure 6.3:

Next we consider the pretzel link P (−n, n+1, n(n+1)) for an integer n ≥ 2, and show the following.

Theorem 6.3. For an integer n ≥ 2, the pretzel link P (−n, n + 1, n(n +

1)) has a minimal diagram admitting only Z-colorings with n2 + n + 3 colors.

Such pretzel links are all Z-colorable by [1] for example. In fact, the 6.1. PRETZEL LINKS 83 determinant of the link P (−n, n + 1, n(n + 1)) is calculated as |(−n) ·

(n + 1) + (−n) · n(n + 1) + (n + 1) · n(n + 1)| = 0.

This theorem can be proved in the same way as for Theorem 6.2.

Proof of Theorem 6.3. Let D be the diagram illustrated in Figure 6.4. It is a minimal diagram of P (−n, n + 1, n(n + 1)) as in the previous cases.

0 0 We set the labels x1, . . . , xn+2, x1, . . . , xn+3, y1, . . . , yn2+n+2 of the arcs of

D as shown in Figure 6.4. Remark that some of the arcs are labelled in duplicate.

Figure 6.4: 84 CHAPTER 6. MINIMAL DIAGRAMS

Suppose that a non-trivial Z-coloring γ on D is given. As shown in

[7], without changing the number of the colors, we may assume that the minimum of the colors for γ is 0, and the arcs colored by 0 cannot cross over the arcs colored by the other colors. Thus, on the diagram D, there

are only two arcs x1 and xn+2 which can be colored by 0. We here assume

that the color of the arc x1 is 0, since the case that the arc xn+2 is colored by 0 can be treated in a the same way.

0 Set the color of x2 as a and the color of x2 as b. Note that both a and b have to be non-zero, for γ is assumed to be non-trivial.

Then, for x1, x2, . . . , xn, xn+1, xn+2 we see from Figure 6.4, we see that

0 0 0 γ(xi) = (i − 1)a for 1 ≤ i ≤ n + 2. Also, for x1(= x1), x2, . . . , xn+2(=

0 0 xn+1), xn+3, we see that γ(xj) = (j − 1)b for 1 ≤ j ≤ n + 3. Since xn+1

0 and xn+2 express the same arc, na = (n + 1)b must hold. Thus we obtain that (a, b) = ((n + 1)t, nt) with some positive integer t, for n and n + 1 are co-prime if n ≥ 2.

0 Now, the color of y1 = x2 is b = nt and the color of y2 = x2 is (n + 6.1. PRETZEL LINKS 85

0 1)t. It follows that, for the arcs y1(= x2), y2(= x2), y3, . . . , yn(n+1)+1(=

0 xn+3), yn(n+1)+2(= xn+2), we see that γ(yk) = (n + k − 1)t for 1 ≤ k ≤ n(n + 1) + 2.

We need to check the compatibility of the colors for the arcs labelled

in duplicate. The colors of yn(n+1)+1 is (n+(n(n+1)+1)−1)t = n(n+2)t,

0 which is equal to the color ((n + 3) − 1)b of xn+3. The colors of yn(n+1)+2 is (n + (n(n + 1) + 2) − 1)t = (n + 1)2t, which is equal to the color

((n + 2) − 1)a of xn+2.

Consequently, after dividing all the colors by t, γ is illustrated by

Figure 6.5. The colors appearing there are {0, n + 1, 2(n + 1), . . . , n(n +

1), (n + 1)2} ∪ {0, n, 2n, . . . , (n + 1)n, (n + 2)n} ∪ {n, n + 1, n + 2, . . . , n + n(n + 1), n + n(n + 1) + 1 = (n + 1)2}.

Note that {n + 1, 2(n + 1), . . . , n(n + 1), (n + 1)2} ⊂ {n, n + 1, n +

2, . . . , n+n(n+1), n+n(n+1)+1 = (n+1)2} and {n, 2n, . . . , (n+1)n, (n+

2)n} ⊂ {n, n + 1, n + 2, . . . , n + n(n + 1) = (n + 2)n, n + n(n + 1) + 1 =

(n + 1)2}. 86 CHAPTER 6. MINIMAL DIAGRAMS

Therefore, there are only mutually distinct colors {0, n, n + 1, n +

2, n + 3, . . . , n + n(n + 1), n + n(n + 1) + 1 = (n + 1)2}, and so, we see

2 mincolZ(D) = 2 + n(n + 1) + 1 = n + n + 3.

Figure 6.5:

6.2 Torus links

In this section, we consider torus links, that is, the links which can be isotoped onto the standardly embedded torus in the 3-sphere. By T (a, b), we mean the torus link running a times meridionally and b times longi- 6.2. TORUS LINKS 87 tudinally. It is known that T (a, b) is Z-colorable if a or b are even.

Theorem 6.4. Let p, q and r be non-zero integers such that |p| ≥ q ≥ 1 and r ≥ 2. Let D be the standard diagram of T (pr, qr) illustrated by

Figure 6.6. And if r is even, the minimal coloring number of D is four.

If r is odd, the minimal coloring number of D is at most five.

Figure 6.6:

Remark that the diagram D has the least number of crossings for the torus link, that is well-known. See [10] for example.

Proof of Theorem 6.4. In the following, we will find a Z-coloring γ on D 88 CHAPTER 6. MINIMAL DIAGRAMS by assigning colors on the arcs of D.

Note that the link has r components running longitudinally with p times twistings meridionally as shown in Figure 6.6. In a local view, we see qr horizontal parallel arcs in D. We divide such arcs into q subfamilies

x1,..., xq as depicted in Figure 6.7.

Figure 6.7:

We first find a local Z-coloring γ on the local diagram shown in Figure

6.7. In the case r is even, we start with setting γ(xi) = (γ(xi,1), γ(xi,2), . . . , γ(xi,r))

= (1, 0,..., 0, 1) for any i. See Figure 6.8. 6.2. TORUS LINKS 89

Figure 6.8:

As illustrated in Figures 6.9, we can extend γ on the arcs in the regions (1) and (q + 1) in Figure 6.8. 90 CHAPTER 6. MINIMAL DIAGRAMS

Figure 6.9:

As illustrated in Figures 6.10, we can extend γ on the arcs in the regions (2), (3),..., (q) in Figure 6.8.

Figure 6.10:

Now, γ can be extended on all the arcs in the region depicted in

Figure 6.11. 6.2. TORUS LINKS 91

Figure 6.11:

Since D is composed of p copies of the local diagram in Figure 6.8, it concludes that D admits a Z-coloring with only four colors 0, 1, 2 and 3.

In the case r is odd and p is even, we also start with setting γ(xi)

= (γ(xi,1), γ(xi,2), . . . , γ(xi,r)) = (1, 0,..., 0, 1) for any i. See Figure 6.8 again. 92 CHAPTER 6. MINIMAL DIAGRAMS

As illustrated in Figures 6.12, we can extend γ on the arcs in the region (1) in Figure 6.8.

Figure 6.12:

As illustrated in Figures 6.13, we can extend γ on the arcs in the regions (2), (3),..., (q) in Figure 6.8.

Figure 6.13: 6.2. TORUS LINKS 93

As illustrated in Figures 6.14, we can extend γ on the arcs in the region (q + 1) in Figure 6.8.

Figure 6.14:

Then, as shown in Figure 6.15 we can extend γ to all the arcs in

Figure 6.8. 94 CHAPTER 6. MINIMAL DIAGRAMS

Figure 6.15:

Without contradicting to the condition of the coloring, we can connect the local diagram in Figure 6.15 with the image of π-rotation. See Figure

6.16. 6.2. TORUS LINKS 95

Figure 6.16:

That is, since p is even, D has a Z-coloring which is composed by connecting p/2 local diagrams illustrated by Figure 6.16.

It concludes that the colors of this Z-coloring are {0, 1, 2, 3, 4}, that is, the Z-coloring is represented by five colors. 96 CHAPTER 6. MINIMAL DIAGRAMS

In the case r is odd and q is even, we start with setting

   (2, 1,..., 1, 0) if i is odd,  γ(xi) =   (0, 1,..., 1, 2) if i is even. 

Then we see the local diagram with a Z-coloring as follows.

Figure 6.17: 6.2. TORUS LINKS 97

As illustrated in Figures 6.18, we can extend γ on the arcs in the regions (1) and (q + 1) in Figure 6.8.

Figure 6.18:

As illustrated in Figure 6.19, we can extend γ on the arcs in the regions (2), (3),..., (q) in Figure 6.8.

Figure 6.19: 98 CHAPTER 6. MINIMAL DIAGRAMS

Now, γ can be extended on all the arcs in Figure 6.20.

Figure 6.20:

Since D is composed of p copies of the local diagram in Figure 6.8, it concludes that D admits a Z-coloring with only four colors 0, 1, 2, 3 and

4.

In the special case of T (3p, 3), we can have the following. 6.2. TORUS LINKS 99

Theorem 6.5. Let p be non-zero integer. Let D be the standard dia- gram of T (3p, 3) illustrated by Figure 6.6. And if p is even, the minimal coloring number of D is five.

Proof. For a torus link T (6, 3), we have the following minimal diagram with a Z-coloring γ with integers 0, a > 0 and b > 0.

Figure 6.21:

The colors of γ is {0, a, 2a, 3a, b, 2b, 3b, 2a − b, 2b − a, 2b + 2a, b + 2a}.

Here we assume that b 6= a. In the case of b = 2a, we see that 2b = 4a and 3b = 6a. In the case b = 3a, we see that 2b = 6a and 3b = 9a. In the case b 6= 2a, 3a, we see that 2a − b 6= 0, a, 2a, 3a, b. That is, if b 6= a,

γ has the colors greater than five. Therefore we obtain a = b in order for 100 CHAPTER 6. MINIMAL DIAGRAMS

γ to have at most five colors.

That is, the diagram with γ is represented in the Figure 6.22.

Figure 6.22:

Here the colors of γ are 0, a, 2a, 3a and 4a. Since a is not 0, the number of the color is five. That is, T (3p, 3) with an even number p has no Z-colorings with four colors. Acknowledgement

I would like to express my gratitude to my thesis’s advisor, Kazuhiro

Ichihara, for his supports and advices. I am grateful to Takuji Naka- mura,Yasutaka Nakanishi and Shin Satoh for useful discussions and ad- vices. In particular in Chapter 3, I have greatly benefited from them.

I am also grateful to Kimihiko Motegi for useful discussions. I wish to thank Ryo Nikkuni, Kanako Oshiro and Chuichiro Hayashi for the lec- tures about the various knowledges of knot theory.

I would like to thank Meiqiao Zhang for useful discussions. I also thank

Akio Kawauchi for giving me the motivation that I consider about par- allels, and to Kouki Taniyama for pointing out the necessity of Lemmas

101 102 CHAPTER 6. MINIMAL DIAGRAMS

5.7. I am also grateful to Ayumu Inoue, Takuji Nakamura and Shin

Satoh for useful discussions and advices. And I would like to thank to the anonymous referee for his/her careful reading and pointing out a gap in our proof of a theorem in [7]. Finally, I’m grateful to all the people who supported me. Bibliography

[1] O. T. Dasbach, D. Futer, E. Kalfagianni, X.-S. Lin, and N. W.

Stoltzfus, Alternating sum formulae for the determinant and other

link invariants, J. Knot Theory Ramifications 19 (2010), no. 6, 765–

782.

[2] J. C. Cha and C. Livingston, LinkInfo: Table of Knot Invariants,

http://www.indiana.edu/~linkinfo, (January 13, 2016).

[3] R. H. Fox, A quick trip through knot theory, in Topology of 3-

manifolds and related topics (Proc. The Univ. of Georgia Institute,

1961), 120–167, Prentice Hall, Englewood Cliffs, NJ.

[4] J. Ge, X. Jin, L.H. Kauffman, P. Lopes, and L. Zhang, Minimal

103 104 BIBLIOGRAPHY

sufficient sets of colors and minimum number of colors, J. Knot

Theory Ramifications 26 (2017), no. 9, 1743008, 16 pp.

[5] F. Harary and L. H. Kauffman, Knots and graphs. I. Arc graphs

and colorings, Adv. in Appl. Math. 22 (1999), no. 3, 312–337.

[6] K. Ichihara and E. Matsudo, A lower bound on minimal number of

colors for links, Kobe J. Math. 33 (2016), no. 1-2, 53–60.

[7] K. Ichihara and E. Matsudo, Minimal coloring number for Z-

colorable links, J. Knot Theory Ramifications 26 (2017), no. 4,

1750018, 23 pp.

[8] K. Ichihara and E. Matsudo, Minimal coloring number on minimal

diagrams for Z-colorable links, Proc. Inst. Nat. Sci., Nihon Univ.53

(2018), 231-237.

[9] L. H. Kauffman and P. Lopes, On the minimum number of colors

for knots, Adv. in Appl. Math. 40 (2008), no. 1, 36–53. BIBLIOGRAPHY 105

[10] A. Kawauchi, Lectures on knot theory, Monograph in Japanese, 2007,

Kyoritsu Shuppan Co. Ltd.

[11] W. B. R. Lickorish, An introduction to knot theory, Graduate Texts

in Mathematics, 175, Springer, New York, 1997.

[12] W. B. R. Lickorish and M. B. Thistlethwaite, Some links with non-

trivial polynomials and their crossing-numbers, Comment. Math.

Helv. 63 (1988), no. 4, 527–539.

[13] P. Lopes and J. Matias, Minimum number of Fox colors for small

primes, J. Knot Theory Ramifications 21 (2012), no. 3, 1250025, 12

pp.

[14] E. Matsudo, Minimal coloring number for Z-colorable links II, to

apper in J. Knot Theory Ramifications, arXiv:1705.07567v3.

[15] E. Matsudo, On the minimal coloring number of even-parallels of

links, to apper in Proc. Inst. Nat. Sci., Nihon Univ. 106 BIBLIOGRAPHY

[16] T. Nakamura, Y. Nakanishi and S. Satoh, The palette graph of a

Fox coloring, Yokohama Math. J. 59 (2013), 91–97.

[17] T. Nakamura, Y. Nakanishi and S. Satoh, On effective 9-colorings

for knots, J. Knot Theory Ramifications 23 (2014), no. 12, 1450059,

15 pp.

[18] K. Oshiro, Any 7-colorable knot can be colored by four colors, J.

Math. Soc. Japan 62 (2010), no. 3, 963–973.

[19] S. Satoh, 5-colored knot diagram with four colors, Osaka J. Math.

46 (2009), no. 4, 939–948.

[20] M. Zhang, X. Jin and Q. Deng, The minimal coloring number of any

non-splittable Z-colorable link is four, J. Knot Theory Ramifications

26 (2017), no. 13, 1750084, 18 pp.