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Minimal Coloring Numbers of Z-Colorable Links

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Minimal Coloring Numbers of Z-Colorable Links 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 Knots, 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 tricolorability is much often used to prove that the trefoil is a non-trivial knot. 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 link 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 : farc of Dg ! 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 Knot theory, 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 L^ = π(L) is called a regular projection of L. We introduce crossing information to L^ 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 Reidemeister move. 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 : farcs of Dg ! 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].
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