Minimal coloring numbers on minimal diagrams of torus links

TL;DR

This paper determines the minimal number of colors needed for non-trivial integer colorings of minimal diagrams of torus links, providing classifications of such colorings and those using only four colors through rack coloring methods.

Contribution

It introduces a complete classification of minimal colorings and four-color colorings of torus links using rack colorings, advancing understanding of link colorings.

Findings

Identifies minimal coloring numbers for torus links.

Classifies all such colorings and four-color colorings.

Uses rack colorings to achieve classifications.

Abstract

We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such $\mathbb{Z}$-colorings by only four colors, which are shown by using rack colorings on link diagrams.

Equations

$$(\gamma (x_{i,1}),\gamma (x_{i,2}),\dots ,\gamma (x_{i,r}))=\{\begin{array}{ll}(2,1,\dots ,1,0)& \text{ if i is odd,}\\ (0,1,\dots ,1,2)& \text{ if i is even.}\end{array}$$

$$r(\mathcal{C})=(\mathcal{C}({\gamma }_1),\dots ,\mathcal{C}({\gamma }_n)).$$

$$\left\{\begin{array}[]{ll}\{(a_{1},\dots,a_{n})\mid w(a_{1}-a_{2}+a_{3}-\cdots-a_{n})=0\}&\text{if $n$ is even,}\\[2.0pt] \{(a,\dots,a)\mid a\in\mathbb{Z}\}&\text{if $n$ is odd and $w$ is odd,}\\[2.0pt] \mathbb{Z}^{n}&\text{if $n$ is odd and $w$ is even,}\end{array}\right.$$

$$A=\left\{({\bm{a}}_{1},\dots,{\bm{a}}_{q})\in(\mathbb{Z}^{r})^{q}\middle|\begin{tabular}[]{l}the assignment of ${\bm{a}}_{1},\dots,{\bm{a}}_{q}\in\mathbb{Z}^{r}$ to ${\bm{x}}_{1},\dots,{\bm{x}}_{q}$\\ defines a $\mathbb{Z}$-coloring of $D$\end{tabular}\right\}.$$

$$A=\left\{\begin{tabular}[]{ll}$\{({\bm{a}},\dots,{\bm{a}})\mid{\bm{a}}\in\mathbb{Z}^{r},\Delta({\bm{a}})=0\}$&if $r$ is even,\\ $\{({\bm{a}},\dots,{\bm{a}})\mid{\bm{a}}\in\mathbb{Z}^{r}\}$&if $r$ is odd and $p$ is even,\\ $\{({\bm{a}},\tau({\bm{a}}),{\bm{a}},\dots,\tau({\bm{a}}))\mid{\bm{a}}\in\mathbb{Z}^{r}\}$&if $r$ is odd and $q$ is even,\\ \end{tabular}\right.$$

$$\left\{\begin{array}[]{ll}0,0,0,0,\dots,0&\text{if $p$ is even and $q$ is odd,}\\ 0,1,0,1,\dots,0&\text{if $p$ is odd and $q$ is even.}\end{array}\right.$$

$$A^{(4)}=\left\{({\bm{a}}_{1},\dots,{\bm{a}}_{q})\in(\mathbb{Z}^{r})^{q}\middle|\begin{tabular}[]{l}the assignment of ${\bm{a}}_{1},\dots,{\bm{a}}_{q}\in\mathbb{Z}^{r}$ to ${\bm{x}}_{1},\dots,{\bm{x}}_{q}$\\ defines a $\mathbb{Z}$-coloring of $D$ with the four colors $\{0,1,2,3\}$\end{tabular}\right\}.$$

$$A^{(4)}=\left\{(\mathbf{a},\dots ,\mathbf{a})|\mathbf{a}\in A_{01}^{(4)}\cup A_{12}^{(4)}\cup A_{23}^{(4)}\right\}\backslash \{(1,\dots ,1),(2,\dots ,2)\},$$

$$A_{01}^{(4)}$$

$$A_{12}^{(4)}$$

$$A_{23}^{(4)}$$

$$z_i=(((x_i\ast y_1)\ast y_2)\cdots )\ast y_n.$$

$$(x_{i})_{i}*_{Q}(y_{i})_{i}=\big{(}x_{i}+2(-x_{n}+x_{n-1}-\cdots+(-1)^{n}x_{1}+(-1)^{n+1}y_{1}+\cdots+y_{n})\big{)}_{i},$$

$$(x_{i}+a)_{i}*_{Q}(x_{i}+b)_{i}=\big{(}x_{i}+a+(1-(-1)^{n})(b-a)\big{)}_{i}=\big{(}x_{i}+(-1)^{n}a+(1-(-1)^{n})b\big{)}_{i},$$

$${\tau }^m(y)\ast {\tau }^n(z)$$

$$={\tau }^{m+1}(y{\ast }_{Q}z)$$

$$\tau({\bm{a}})=\big{(}(-1)^{n}a_{i}+2(a_{n}-a_{n-1}+\cdots+(-1)^{n-1}a_{1})\big{)}_{i}.$$

$$\tau^{w}({\bm{a}})=\big{(}a_{i}+2w(a_{n}-a_{n-1}+\cdots-a_{1})\big{)}_{i}.$$

Full text

The first author is partially supported by JSPS KAKENHI Grant Number JP18K03287.

The second author was partially supported by JSPS KAKENHI Grant Number JP16J01183.

1. Introduction

This is a continuation of the study of $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links given in [6, Section 3] by the first and the third named authors.

Previously, in [5], as a generalization to the well-known Fox’s coloring originally introduced in [3], they defined a $\mathbb{Z}$-coloring for a link in the 3-space as follows. A map $\gamma:\{$arcs of $D\}\rightarrow\mathbb{Z}$ for a regular diagram $D$ of a link is called a $\mathbb{Z}$-coloring if it satisfies the condition $2\gamma(a)=\gamma(b)+\gamma(c)$ at each crossing of $D$ with the over arc $a$ and the under arcs $b$ and $c$. We say that a link is $\mathbb{Z}$-colorable if it has a diagram admitting a non-trivial $\mathbb{Z}$-coloring, i.e., there are at least two distinct colors on the diagram. We remark that a link $L$ is $\mathbb{Z}$-colorable if and only if the determinant $\det(L)$ of $L$ equals [math]. Since the determinant of any knot (single component link) is shown to be an odd integer, any knot is not $\mathbb{Z}$-colorable.

There are several studies on the minimal coloring number (i.e., the minimal number of the colors) of Fox colorings on knots and links; some upper and lower bounds have been obtained. In the same line, it was shown in [5, Theorem 3.1] that the minimal coloring number $mincol_{\mathbb{Z}}(L)$ of a non-splittable $\mathbb{Z}$-colorable link $L$ is at least four. (Note that $mincol_{\mathbb{Z}}(L)=2$ for any splittable link $L$.) However, in contrast to the case of the Fox coloring, it was proved that $mincol_{\mathbb{Z}}(L)=4$ for any non-splittable $\mathbb{Z}$-colorable link $L$, by the third author in [8], and independently by Meiqiao Zhang, Xian’an Jin, and Qingying Deng in [10].

Thereafter, in [6], the first and third authors consider the minimal coloring number $mincol_{\mathbb{Z}}(D)$ of a minimal diagram $D$ of a $\mathbb{Z}$-colorable link. (A minimal diagram is a diagram representing the link with least number of crossings.)

In particular, in [6, Section 3], they consider torus links, that is, the links which can be isotoped onto the standardly embedded torus in the $3$-space. By $T(a,b)$, we mean the torus link running $a$ times meridionally and $b$ times longitudinally. It is known that $T(a,b)$ is $\mathbb{Z}$-colorable if $a$ or $b$ is even. Actually, it is shown in [6, Theorem 1.3] that $mincol_{\mathbb{Z}}(D)=4$ for the standard diagram $D$ of $T(pn,n)$ illustrated by Figure 1 with $n>2$, even and $p\neq 0$. Extending this, in this paper, we show the following:

Theorem 1.1**.**

Let $p,q,$ and $r$ be non-zero integers such that $p$ and $q$ are relatively prime, $|p|\geq q\geq 1,$ and $r\geq 2$. Let $D$ be the standard diagram of $T(pr,qr)$ illustrated by Figure 1. Suppose that $T(pr,qr)$ is $\mathbb{Z}$-colorable, or, equivalently, $pr$ or $qr$ is even. Then, $mincol_{\mathbb{Z}}(D)=4$ if $r$ is even, and $mincol_{\mathbb{Z}}(D)=5$ if $r$ is odd.

We remark that the diagram $D$ has the least number of crossings for the torus link. See [4] for example. Also if $r=1$, then the link becomes a knot, which is not $\mathbb{Z}$-colorable. (Actually $r$ coincides with the number of components of the torus link.) The theorem above is proved in Section 2 (when $r$ is even) and Section 3 (when $r$ is odd).

We also include complete classifications of all the $\mathbb{Z}$-colorings on the standard diagram of $T(a,b)$ (Proposition 4.3) and of all the $\mathbb{Z}$-colorings by only four colors of $T(a,b)$ (Proposition 4.4) in Section 4. The key of our proof of the propositions is to use rack colorings on link diagrams. A theorem used to prove the propositions, which can be of interest independently, is proved in Appendix. That part is essentially based on the master thesis [7] of the second author.

2. Four colors case

In this section, we prove the following, showing the first assertion of Theorem 1.1.

Theorem 2.1**.**

Let $p,q,$ and $r$ be non-zero integers such that $p$ and $q$ are relatively prime, $|p|\geq q\geq 1,$ and $r\geq 2$. Let $D$ be the standard diagram of $T(pr,qr)$ illustrated by Figure 1. Suppose that $T(pr,qr)$ is $\mathbb{Z}$-colorable, or, equivalently, $pr$ or $qr$ is even. Then, $mincol_{\mathbb{Z}}(D)=4$ if $r$ is even.

Proof.

We will find a $\mathbb{Z}$-coloring $\gamma$ with only four colors on $D$ by assigning colors on the arcs of $D$.

Note that the link has $r$ components each of which runs longitudinally $q$ times and twists meridionally $p$ times as shown in Figure 1. In a local view, we see $qr$ horizontal parallel arcs in $D$. We divide such arcs into $q$ subfamilies $\mathbf{x}_{1},\dots,\mathbf{x}_{q}$ as depicted in Figure 2 (left).

We first find a local $\mathbb{Z}$-coloring $\gamma$ on the local diagram shown in Figure 2 (left). Let us start with setting $(\gamma(x_{i,1}),\gamma(x_{i,2}),\dots,\gamma(x_{i,r}))=(1,0,\dots,0,1)$ for any $i$. See Figure 2 (right).

Since $r$ is assumed to be even, as illustrated in Figures 3 (left), we can extend $\gamma$ on the arcs in the regions $(1)$ and $(q+1)$ in Figure 2 (right), and, as illustrated in Figures 3 (right), we can extend $\gamma$ on the arcs in the regions $(2),(3),\dots,(q)$ in Figure 2 (right). Thus $\gamma$ can be extended on all the arcs in the local diagram as depicted in Figure 4.

Since $D$ is composed of $p$ copies of the local diagram in Figure 2 (left), it concludes that $D$ admits a $\mathbb{Z}$-coloring with only four colors [math], $1$, $2$, and $3$.

Since the torus link is non-splittable, $mincol_{\mathbb{Z}}(D)$ must be at least four by [5, Theorem 3.1], and we conclude that $mincol_{\mathbb{Z}}(D)=4$. ∎

3. Five colors case

In this section, we prove the following, which asserts the second half of the statement of Theorem 1.1.

Theorem 3.1**.**

Let $p,q,$ and $r$ be non-zero integers such that $p$ and $q$ are relatively prime, $|p|\geq q\geq 1$, and $r\geq 2$. Let $D$ be the standard diagram of $T(pr,qr)$ illustrated by Figure 1. Suppose that $T(pr,qr)$ is $\mathbb{Z}$-colorable, or, equivalently, $pr$ or $qr$ is even. Then $mincol_{\mathbb{Z}}(D)=5$ if $r$ is odd.

To prove this, we recall the following lemma, which was obtained in [5].

Lemma 3.2**.**

For a $\mathbb{Z}$-coloring $\gamma$ with $0=\min{\rm Im}\,\gamma$, if an over arc at a crossing is colored by [math], then the under arcs at the crossing are colored by [math]. For a $\mathbb{Z}$-coloring $\gamma$ with $M=\max{\rm Im}\,\gamma$, if an over arc at a crossing is colored by $M$, then the under arcs at the crossing are colored by $M$.

Proof of Theorem 3.1.

We first show that the minimal coloring number $mincol_{\mathbb{Z}}(D)$ of $D$ is at most five, i.e., $mincol_{\mathbb{Z}}(D)\leq 5$.

In the following, we will find a $\mathbb{Z}$-coloring $\gamma$ with five colors on $D$ by assigning colors on the arcs of $D$. In the same way as in the proof of Theorem 2.1, we find a local $\mathbb{Z}$-coloring $\gamma$ on the local diagram shown in Figure 2 (left), and extend it to whole the diagram. Note that if $r$ is odd and $T(pr,qr)$ is $\mathbb{Z}$-colorable, then either $p$ or $q$ must be even.

First, suppose that $p$ is even. In this case, we start with setting $(\gamma(x_{i,1}),\gamma(x_{i,2}),\dots,\gamma(x_{i,r}))=(1,0,\dots,0,1)$ for any $i$. See Figure 2 (right) again. As illustrated in Figure 5 (left) and (right), we can extend $\gamma$ on the arcs in the regions $(1)$ and $(q+1)$ in Figure 2 (right).

As illustrated in Figure 6 (left), we can extend $\gamma$ on the arcs in the regions $(2),(3),\dots,(q)$ in Figure 2 (right). Then, as shown in Figure 6 (right), we can extend $\gamma$ to all the arcs in Figure 2 (right).

Without contradicting to the condition of the coloring, we can connect the local diagram in Figure 6 (right) with the image of $\pi$-rotation. See Figure 7.

That is, since $p$ is assumed to be even, $D$ admits a $\mathbb{Z}$-coloring which is composed by connecting $p/2$ local diagrams illustrated by Figure 7. It concludes that the colors of this $\mathbb{Z}$-coloring are $\{0,1,2,3,4\}$, that is, the $\mathbb{Z}$-coloring is represented by five colors.

Next, suppose that $r$ is odd and $q$ is even. In this case, we start with setting

[TABLE]

See Figure 8 (left).

As illustrated in Figure 8 (right), we can extend $\gamma$ on the arcs in the regions $(1)$ and $(q+1)$ in Figure 2 (right). And, as illustrated in Figure 9, we can extend $\gamma$ on the arcs in the regions $(2),(3),\dots,(q)$ in Figure 2.

Now, $\gamma$ can be extended on all the arcs in Figure 10.

Since $D$ is composed of $p$ copies of the local diagram in Figure 2, it concludes that $D$ admits a $\mathbb{Z}$-coloring with only five colors [math], $1$, $2$, $3$, and $4$.

Consequently, we obtain that $mincol_{\mathbb{Z}}(D)\leq 5$.

We next show that the minimal coloring number $mincol_{\mathbb{Z}}(D)$ of $D$ is at least five, i.e., $mincol_{\mathbb{Z}}(D)\geq 5$.

Suppose for a contradiction that the diagram $D$ as shown in Figure 1 admits a non-trivial $\mathbb{Z}$-coloring $\gamma$ with only four colors. By [5, Theorem 3.2], we may assume that the image of $\gamma$ is $\{0,1,2,3\}$. By Lemma 3.2, we see that the over arcs are colored by $1$ or $2$, otherwise $\gamma$ have to be trivial. Thus there exist only three ways to color a crossing:

Here we see that, in a component including an arc colored by 1 or 3 (resp. 0 or 2), the arcs are always colored by odd (resp. even) numbers, by the condition of the $\mathbb{Z}$-coloring.

Then, since the number of the component is the odd number $r$, either the number of the components colored by odd numbers or that by even numbers is odd. Since the linear function $X\mapsto-X+3$ on $\mathbb{Z}$ switches these two cases, we only have to consider the former case; then the number of the over arcs colored by $1$ is odd in the $r$ parallel over arcs as in Figure 12.

In the case $q=1$, we consider the $r$ parallel arcs $\mathbf{x}=\{x_{1},\dots,x_{r}\}$ as shown in Figure 13.

The colors on $\mathbf{x}$ is represented by $\gamma(\mathbf{x})=(\gamma(x_{1}),\dots,\gamma(x_{r}))$.

By the assumption that $\gamma$ is a non-trivial $\mathbb{Z}$-coloring, the diagram has an over arc colored by $2$. Hence we can label $\mathbf{x}$ to have $x_{r}$ be colored by $2$. That is, we consider the case the colors of $\mathbf{x}$ is $\gamma(\mathbf{x})=(a_{1},a_{2},\dots,a_{r-1},2)$,

The arc $x_{r}$ turns into $x^{\prime}_{r}$ by passing through the odd arcs colored by $1$ and even arcs colored by $2$. Here, as the diagram has exactly four colors $0,1,2,3$, the arc $x^{\prime}_{r}$ is colored by [math]. That is, there exists an over arc colored by [math] as shown in Figure 14.

This is a contradiction; the colors on over arcs are $1$ or $2$.

In the case $q\geq 2$, We divide $qr$ parallel arcs into $q$ subfamilies $\mathbf{x}_{1},\dots,\mathbf{x}_{q}$ and $\mathbf{y}_{1},\dots,\mathbf{y}_{q}$ as depicted in Figure 12. The colors on $\mathbf{x}_{i}=(x_{i,1},\dots,x_{i,r})$ are represented by $\gamma(\mathbf{x}_{i})=(\gamma(x_{i,1}),\dots,\gamma(x_{i,r}))$.

By the condition of a $\mathbb{Z}$-coloring, the colors on $\mathbf{y}_{i}$ are expressed by using a linear function $f$ as $\gamma(\mathbf{y}_{i})=\{f(\gamma(x_{i,1})),\dots,f(\gamma(x_{i,r}))\}$ with $i=2,3,\dots,q$. Then we see $f(0)=2$ and $f(2)=0$, as the number of the over arcs colored by $1$ is odd in the $r$ over arcs in Figure 12.

Here we obtain $f(X)=-X+2$ and see $f(3)=-1$. That is, if there exists an arc colored by $3$ in any parallel arcs $\mathbf{x}_{i}$ ($i=2,\dots,q$), there exists an arc colored by $-1$ in the parallel arcs $\mathbf{y}_{i}$. It is a contradiction since the image of $\gamma$ is $\{0,1,2,3\}$.

By the assumption, the diagram has an arc colored by $3$. If there exist no arcs colored by $3$ in parallel arcs $\mathbf{x}_{i}$, we relabel the arcs to have the arc colored by $3$ be one of parallel arcs. Furthermore, if there exist an arc colored by $3$ in $\mathbf{x}_{1}$, we see the outside of the local diagram as shown in Figure 12 and we relabel to have $\mathbf{x}_{1}$ as $\mathbf{y}_{1}$. Since $f$ is an involution, this completes the theorem.

∎

4. $\mathbb{Z}$-colorings of torus link diagrams

In this section, we give complete classifications of all $\mathbb{Z}$-colorings of the standard diagram of $T(a,b)$ (Proposition 4.3) and of all $\mathbb{Z}$-colorings by only four colors of $T(a,b)$ (Proposition 4.4).

To achieve these, we prepare a theorem on $\mathbb{Z}$-colorings of $n$-parallels of knots. For a knot diagram $D$, we obtain another diagram $D^{(n)}$ by replacing the string with $n$ parallel copies of it; we call $D^{(n)}$ the $n$-parallel of $D$. Remark that $D^{(n)}$ represents the $(nw,n)$-cable link of the knot represented by $D$, where $w$ is the writhe of $D$. In the following, we denote the set of $\mathbb{Z}$-colorings of a link diagram $D$ by ${\rm Col}_{\mathbb{Z}}(D)$.

Theorem 4.1**.**

Let $D$ be an oriented knot diagram and $D^{(n)}$ the $n$-parallel of $D$. We fix any arc of $D$ and let $\gamma_{1},\dots,\gamma_{n}$ be the corresponding $n$ arcs of $D^{(n)}$. We define a homomorphism $r:{\rm Col}_{\mathbb{Z}}(D^{(n)})\to\mathbb{Z}^{n}$ as

[TABLE]

Then, $r$ is injective and the image ${\rm Im}\,r$ of $r$ is equal to

[TABLE]

where $w$ is the writhe of $D$.

Remark 4.2*.*

Theorem 4.1 states that the colors $a_{1},\dots,a_{n}$ of the $n$-paralleled arcs of an arc $\gamma$ determine the whole coloring, especially the colors $a^{\prime}_{1},\dots,a^{\prime}_{n}$ of the $n$-parallel of another arc $\gamma^{\prime}$. As we see in the proof, we can calculate $a^{\prime}_{1},\dots,a^{\prime}_{n}$ from a cyclic-rack coloring of $D$. For example, if $n$ is even and $w\neq 0$, we always have $a^{\prime}_{i}=a_{i}$ for any $i$. If $n$ is odd and $w$ is even, we track the string of $D$ from $\gamma$ to $\gamma^{\prime}$ and count the number of times of passing under arcs; if it is even, then $a^{\prime}_{i}=a_{i}$, and otherwise, $a^{\prime}_{i}=-a_{i}+2(a_{1}-a_{2}+\cdots+a_{n})$.

We include a proof of Theorem 4.1 in Appendix, for it is rather independent from the other contents of the paper.

4.1. Determining $\mathbb{Z}$-colorings

Let $B(a,b)$ denote the braid (diagram) illustrated in Figure 16. We can regard the torus link $T(a,b)$ as the closure of $B(a,b)$. Then, an assignment of colors to the $b$ left ends of $B(a,b)$ determines a $\mathbb{Z}$-coloring of $B(a,b)$, and if the resulting colors of the right ends coincide with the left colors, it gives a $\mathbb{Z}$-coloring of the standard diagram of $T(a,b)$.

Let $p,q$, and $r$ be nonzero integers such that $|p|\geq q\geq 1$ and $r\geq 2$, and assume that $p$ and $q$ are mutually prime. We divide the $qr$ arcs of the left end of $B(pr,qr)$ into $q$ subfamilies ${\bm{x}}_{1},\dots,{\bm{x}}_{q}$ as depicted in Figure 2 (left). Let $A$ be the set of the $qr$-tuples of integers which give a $\mathbb{Z}$-coloring of the standard diagram $D$ of $T(pr,qr)$, i.e.,

[TABLE]

The following proposition describes the coloring of torus links:

Proposition 4.3**.**

We have

[TABLE]

where $\Delta({\bm{a}})=a_{1}-a_{2}+\cdots+(-1)^{r}a_{r}\in\mathbb{Z}$ and $\tau({\bm{a}})=(-a_{i}+2\Delta({\bm{a}}))_{i}\in\mathbb{Z}^{r}$ for ${\bm{a}}=(a_{1},\dots,a_{r})\in\mathbb{Z}^{r}.$

Proof.

Let $B_{0}$ be the tangle diagram depicted in Figure 17 and $B_{0}^{(r)}$ the $r$-parallel of $B_{0}$. We denote the closures of $B_{0}$ and $B_{0}^{(r)}$ by $D_{0}$ and $D_{0}^{(r)}$, respectively. Since $D_{0}$ represents the knot $T(p,q)$, we can apply Theorem 4.1 to $D_{0}$ and its $r$-parallel $D_{0}^{(r)}$ to determine the colorings of $D_{0}^{(r)}$. Furthermore, we should remark that $B_{0}^{(r)}$ and $B(pr,qr)$ are isotopic; we find the colorings of $D$ from those of $D_{0}^{(r)}$, using an isotopy which takes $D_{0}^{(r)}$ to $D$ and fixes the arcs $x_{i,j}$ ($1\leq i\leq q,1\leq j\leq r$) and their colors.

By Theorem 4.1 (and Remark 4.2), a coloring of $D_{0}^{(r)}$ is determined by the color ${\bm{a}}\in\mathbb{Z}^{r}$ of $r$ arcs ${\bm{x}}_{1}$, and the whole coloring is found from a cyclic-rack coloring of $D_{0}$. If $r$ is even, the color ${\bm{a}}$ has to satisfy the condition $\Delta({\bm{a}})=0$ and then the color of ${\bm{x}}_{i}$ is equal to ${\bm{a}}$ for any $i$. In the other cases, we can choose any ${\bm{a}}\in\mathbb{Z}^{r}$, and we consider the cyclic rack $C_{2}$ of order $2$ to examine the whole coloring. A brief calculation finds a $C_{2}$-coloring of $D_{0}$, which colors the left $q$ arcs

[TABLE]

In terms of Remark 4.2, a color $0\in C_{2}$ corresponds to “even” and $1\in C_{1}$ to “odd”; as in the remark, we associate ${\bm{a}}$ to the $r$-paralleled arcs of an arc if it has a color [math], and associate $\tau({\bm{a}})$ otherwise. This completes the proposition. ∎

4.2. Coloring with four colors

Let $p,q,$ and $r$ be non-zero integers such that $p$ and $q$ are relatively prime, $|p|\geq q\geq 1$, and $r\geq 2$. By Theorem 1.1, the standard diagram $D$ of $T(pr,qr)$ admits a $\mathbb{Z}$-coloring $\gamma$ with four colors if and only if $r$ is even. In this case, it is sufficient to consider the case where ${\rm Im}\,\gamma=\{0,1,2,3\}$ by [5, Theorem 3.2]. Let $A^{(4)}$ be the set of the $qr$-tuples which give such $\mathbb{Z}$-colorings:

[TABLE]

Here, we regard $D$ as the closure of $B(pr,qr)$ and denote the subfamilies of the left $qr$ arcs by ${\bm{x}}_{1},\dots,{\bm{x}}_{q}$ as shown in Figure 2 (left).

Proposition 4.4**.**

We have

[TABLE]

where

[TABLE]

Proof.

By Proposition 4.3, a member of $A^{(4)}$ is expressed as $({\bm{a}},\dots,{\bm{a}})$, where ${\bm{a}}=(a_{1},\dots,a_{r})$ is an $r$-tuple of integers with $\Delta({\bm{a}})=a_{1}-a_{2}+\dots-a_{r}$ equal to [math]. We have to show that ${\bm{a}}\in A^{(4)}_{01}\cup A^{(4)}_{12}\cup A^{(4)}_{23}$.

Let $S$ be the set $\{a_{1},\dots,a_{r}\}$. We first assert that $S=\{0,1\},\{1,2\},$ or $\{2,3\}$. To show $S\not\supset\{0,2\}$, we suppose that $\{a_{i},a_{j}\}=\{0,2\}\;(i<j)$. The arc $x_{1,i}$ goes under the $i-1$ arcs to become an over arc $x^{\prime}_{i}$ with color $a^{\prime}_{i}$, whereas $x_{1,j}$ goes under the same arcs to become $x^{\prime}_{j}$ with color $a^{\prime}_{j}$ and then passes under $x^{\prime}_{i}$. Here, we should remark that $|a^{\prime}_{i}-a^{\prime}_{j}|=2$ since $x_{1,i}$ and $x_{1,j}$ goes under the same arcs, but such a crossing is not allowed, as seen in the proof of Theorem 3.1. Thus we have $S\not\supset\{0,2\}$, and in the same way, we can check that $S\not\supset\{1,3\},\{0,3\}$. Therefore $S$ is a set of two consecutive integers, as asserted.

In the case where $S=\{1,2\}$, we see that $a_{1}=a_{2}$; otherwise, the color $-a_{2}+2a_{1}$ of the over arc next to $x_{1,1}$ would not be $1$ or $2$. After two twists, the colors of the arcs shift cyclically to be $(a_{3},a_{4},\dots,a_{2})$ since passing under two strings with a same color does not change the color. Then, the argument above shows that $a_{3}=a_{4}$. Repeating this, we find $a_{2i-1}=a_{2i}$ for $i=1,\dots,r/2$, i.e., ${\bm{a}}\in A^{(4)}_{12}$.

In the case where $S=\{0,1\}$ (resp. $\{2,3\}$), the colors $a_{1}$ and $a_{r}$ of over arcs have to be $1$ (resp. $2$). After $x_{1,1}$ comes to the bottom, the colors of the arcs are $(-a_{2}+2a_{1},-a_{3}+2a_{1},\dots,-a_{r}+2a_{1},a_{1})$, and this is also a member of $A^{(4)}$. Since the set $\{a_{1},-a_{2}+2a_{1},,\dots,-a_{r}+2a_{1}\}$ is equal to $\{1,2\}$, we have $a_{2i}=a_{2i+1}\;(i=1,\dots,r/2-1)$ as shown above. This means that ${\bm{a}}\in A^{(4)}_{01}$ (resp. $A^{(4)}_{23}$).

Let $A^{\prime}$ be the set $\left\{({\bm{a}},\dots,{\bm{a}})\,\middle|\,{\bm{a}}\in A^{(4)}_{01}\cup A^{(4)}_{12}\cup A^{(4)}_{23}\right\}\backslash\{(1,\dots,1),(2,\dots,2)\}$. We have proven that $A^{(4)}\subset A^{\prime}$ as above. A brief calculation shows that $\Delta({\bm{a}})=0$ for ${\bm{a}}\in A^{(4)}_{01}\cup A^{(4)}_{12}\cup A^{(4)}_{23}$ and hence $({\bm{a}},\dots,{\bm{a}})\in A^{\prime}$ gives a $\mathbb{Z}$-coloring of $D$. To see that every element of $A^{\prime}$ actually defines a four-color coloring, it is sufficient to show that for any $(a_{1},\dots,a_{qr})\in A^{\prime}$, $(-a_{2}+2a_{1},\dots,-a_{qr}+2a_{1},a_{1})$ is also a member of $A^{\prime}$; this means that “one twist preserves $A^{\prime}$”, and then implies that there are only four colors in the resultant $\mathbb{Z}$-coloring. This is verified by the definitions of $A^{(4)}_{01},A^{(4)}_{12}$, and $A^{(4)}_{23}$. Thus we have $A^{\prime}\subset A^{(4)}$ as required. ∎

Appendix A $\mathbb{Z}$-coloring of $n$-parallels of knots

In this appendix, we give a proof of Theorem 4.1. In this proof, we use ideas of racks and quandles. For definitions of racks, quandles, and rack (quandle) colorings, see [2] for example. For a rack $R$, we denote the set of $R$-colorings of an oriented knot diagram $D$ by ${\rm Col}_{R}(D)$. For example, the set $\mathbb{Z}$ equipped with the binary operation $*$ defined by $a*b=2b-a$ is a rack (it is in fact a quandle), and then ${\rm Col}_{\mathbb{Z}}(D)$ is the set of $\mathbb{Z}$-colorings as in Section 4.

We denote the automorphism group of a rack $R=(R,*)$ by ${\rm Aut}(R,*)$. For each $a\in R$, the map $\bullet*a:R\ni x\mapsto x*a\in R$ is by definition is an automorphism of $R$, and then the subgroup of ${\rm Aut}(R,*)$ generated by $\bullet*a\;(a\in R)$ is called the inner automorphism group and denoted by ${\rm Inn}(R,*)$. We say that $R$ is (algebraically) connected if the action of ${\rm Inn}(R,*)$ on $R$ is transitive.

Furthermore, we should recall the quandles and kink maps associated to racks. The associated quandle $R_{Q}$ of a rack $R=(R,*)$ is the pair $(R,*_{Q})$, where the binary operation $*_{Q}$ is defined by $x*_{Q}y=(x\mathbin{\bar{*}}x)*y$ $(x,y\in R)$, and the associated kink map $\tau$ is defined by $\tau(x)=x*x$ $(x\in R)$. A brief calculation shows that $R_{Q}$ is a quandle and $\tau$ is a kink map of $R$ (see, e.g., [1]), i.e., $\tau$ is an automorphism of $R$ and for any $x,y\in R$ we have $x*\tau(y)=x*y$; this implies that $\tau$ is also a kink map of $R_{Q}$. We remark that the quandle $R_{Q}$ is equal to $R$ as a set, and an automorphism of $R$ is also an automorphism of $R_{Q}$; we may regard ${\rm Aut}(R,*)$ and ${\rm Inn}(R,*)$ as subgroups of ${\rm Aut}(R_{Q})$.

To show the theorem, we introduce a rack $\mathbb{Z}^{n}_{R}$ as follows (this is due to [9]): for ${\bm{x}}=(x_{1},\dots,x_{n}),{\bm{y}}=(y_{1},\dots,y_{n})\in\mathbb{Z}^{n},$ we put ${\bm{x}}*_{R}{\bm{y}}={\bm{z}}=(z_{1},\dots,z_{n})$, where

[TABLE]

By a brief calculation, we can check that $\mathbb{Z}^{n}_{R}=(\mathbb{Z}^{n},*_{R})$ is a rack.

In the proof of Theorem 4.1 below, we examine the $\mathbb{Z}^{n}_{R}$-colorings of the knot diagram $D$. Since the fundamental rack of a framed knot is connected, the image of a coloring of $D$ by a rack $R$ is contained in a connected subrack of $R$; if $R$ is decomposed into the maximal connected subracks $R_{\lambda}\;(\lambda\in\Lambda)$ (for the existence and uniqueness of the decomposition, see, e.g., [1]), we have ${\rm Col}_{R}(D)=\bigsqcup_{\lambda\in\Lambda}{\rm Col}_{R_{\lambda}}(D)$. The following lemma describes the subracks $R_{\lambda}$ when $R=\mathbb{Z}^{n}_{R}$.

Lemma A.1**.**

Each maximal connected subrack of $\mathbb{Z}^{n}_{R}$ is a cyclic rack.

Recall that a cyclic rack $C_{k}$ ($k\in\mathbb{Z}_{\geq 0}$) is a cyclic group $\mathbb{Z}/k\mathbb{Z}$ with the binary operation $*$ defined by $a*b=a+1$ ($a,b\in\mathbb{Z}/k\mathbb{Z}$). A rack isomorphic to a cyclic rack is also called a cyclic rack.

Proof of Lemma A.1.

Let $\mathbb{Z}^{n}_{Q}=(\mathbb{Z}^{n},*_{Q})$ be the associated quandle of $\mathbb{Z}^{n}_{R}$ and $\tau$ the associated kink map. We first claim that each maximal connected subquandle of $\mathbb{Z}^{n}_{Q}$ is the trivial quandle of order $1$. In fact, a brief calculation shows that

[TABLE]

for $(x_{i})_{i},(y_{i})_{i}\in\mathbb{Z}^{n}_{Q}.$ This shows that the orbit of $(x_{i})_{i}\in\mathbb{Z}^{n}_{Q}$ under the action of ${\rm Inn}(\mathbb{Z}^{n}_{Q})$ is included in $\{(x_{i}+a)_{i}\mid a\in\mathbb{Z}\}.$ Furthermore, since

[TABLE]

the quandle $\{(x_{i}+a)_{i}\mid a\in\mathbb{Z}\}$ is isomorphic to $\mathbb{Z}$ or a trivial quandle. In either case, each maximal connected subquandle is the trivial quandle of order $1$. The lemma follows from this claim and Lemma A.2 below. ∎

Lemma A.2**.**

Let $R=(R,*)$ be a rack, $R_{Q}=(R,*_{Q})$ the associated quandle, and $\tau$ the associated kink map. We denote the maximal connected subrack of $R$ (resp. $R_{Q}$) containing $x\in R$ (resp. $R_{Q}$) by $\mathcal{M}_{x}$ (resp. $\mathcal{M}^{Q}_{x}$). Then we have $\mathcal{M}_{x}=\bigcup_{m\in\mathbb{Z}}\tau^{m}(\mathcal{M}^{Q}_{x}).$

Proof.

We set $\mathcal{M}^{\prime}_{x}=\bigcup_{m\in\mathbb{Z}}\tau^{m}(\mathcal{M}^{Q}_{x}).$ We have

[TABLE]

and similarly $\tau^{m}(y)\mathbin{*}\tau^{n}(z)=\tau^{m-1}(y\mathbin{\overline{*_{Q}}}z)$ for any $y,z\in R$ (especially for $y,z\in\mathcal{M}^{Q}_{x}$) and $m,n\in\mathbb{Z}$, $\mathcal{M}^{\prime}_{x}$ is a connected subrack of $R$; hence we have $\mathcal{M}^{\prime}_{x}\subset\mathcal{M}_{x}$. Furthermore, by the definitions of $*_{Q}$ and $\tau$, ${\rm Inn}(\mathcal{M}_{x},*)$ is contained in the subgroup of ${\rm Aut}(\mathcal{M}_{x},*_{Q})$ generated by ${\rm Inn}(\mathcal{M}_{x},*_{Q})$ and $\tau$. Since $\tau$ is central in this subgroup (recall that $\tau$ is a kink map of $(\mathcal{M}_{x},*_{Q})$, we have $\mathcal{M}_{x}=\bigcup_{m}\tau^{m}(\mathcal{M}^{\prime Q}_{x})$, where we set $\mathcal{M}^{\prime Q}_{x}={\rm Inn}(\mathcal{M}_{x},*_{Q})\cdot x$, i.e., the orbit of $x$ under the action of ${\rm Inn}(\mathcal{M}_{x},*_{Q})$. Here, the inner automorphism group ${\rm Inn}(\mathcal{M}_{x},*_{Q})$ is by definition generated by $\bullet*_{Q}y$ ($y\in\mathcal{M}_{x}$), but for each $y\in\mathcal{M}_{x}=\bigcup_{m}\tau^{m}(\mathcal{M}^{\prime Q}_{x})$ there exists $y^{\prime}\in\mathcal{M}^{\prime Q}_{x}$ such that $\bullet*_{Q}y=\bullet*_{Q}y^{\prime}$; in fact, if $y\in\tau^{m}(\mathcal{M}^{\prime Q}_{x})$, we can take $\tau^{-m}(y)$ as $y^{\prime}$. Thus, $\mathcal{M}^{\prime Q}_{x}={\rm Inn}(\mathcal{M}^{\prime Q}_{x},*_{Q})\cdot x$, which implies that $\mathcal{M}^{\prime Q}_{x}$ is a connected subquandle. Then we have $\mathcal{M}^{\prime Q}_{x}\subset\mathcal{M}^{Q}_{x}$ and hence $\mathcal{M}_{x}=\bigcup_{m}\tau^{m}(\mathcal{M}^{\prime Q}_{x})\subset\bigcup_{m}\tau^{m}(\mathcal{M}^{Q}_{x})=\mathcal{M}^{\prime}_{x}$, as required. ∎

Proof of Theorem 4.1.

Given a $\mathbb{Z}^{n}_{R}$-coloring $\mathcal{C}^{(n)}=(\mathcal{C}_{1},\dots,\mathcal{C}_{n}):\{\text{arcs}\}\to\mathbb{Z}^{n}_{R}$ on $D$, we put colors $\mathcal{C}_{1}(\alpha),\dots,\mathcal{C}_{n}(\alpha)$ to the $n$ arcs of $D^{(n)}$ corresponding to each arc $\alpha$ of $D$. This is uniquely extended to be a whole coloring on $D^{(n)}$ (see Figure 18). Conversely, a $\mathbb{Z}$-coloring $\mathcal{C}$ of $D^{(n)}$ defines a $\mathbb{Z}^{n}_{R}$ coloring: we associate $(\mathcal{C}(\alpha_{1}),\dots,\mathcal{C}(\alpha_{n}))$ to an arc $\alpha$ of $D$, where $\alpha_{1},\dots,\alpha_{n}$ are the $n$ arcs of $D^{(n)}$ corresponding to $\alpha$; by the definition of $*_{R}$, this defines a $\mathbb{Z}^{n}_{R}$-coloring. Thus, we have a bijection between ${\rm Col}_{\mathbb{Z}^{n}}(D)$ and ${\rm Col}_{\mathbb{Z}}(D^{(n)})$. In the following, we identify these two sets by this bijection.

By Lemma A.1, any $\mathbb{Z}^{n}_{R}$-coloring on $D$ is a coloring by a subrack, which is a cyclic rack. Since we only have to consider cyclic-rack colorings, a given color ${\bm{a}}\in\mathbb{Z}^{n}_{R}$ on the fixed arc $\gamma$ uniquely determines the colors of the other arcs successively, and it defines a whole coloring if and only if they accords when we go back to $\gamma$, i.e., $\tau^{w}({\bm{a}})={\bm{a}},$ where $\tau$ is the associated kink map. Thus, $r$ is injective and ${\rm Im}\,r=\{{\bm{a}}\in\mathbb{Z}^{n}\mid\tau^{w}({\bm{a}})={\bm{a}}\}.$

By a concrete calculation we find that

[TABLE]

If $n$ is even, this shows that

[TABLE]

Then $\tau^{w}({\bm{a}})={\bm{a}}$ if and only if $w(a_{1}-a_{2}+\cdots-a_{n})=0$, as required. Next, suppose that $n$ is odd. In this case, a brief calculation shows that $\tau^{2}={\rm id}_{\mathbb{Z}^{n}}$. Then, if $w$ is even, the condition $\tau^{w}({\bm{a}})={\bm{a}}$ is always satisfied. If $w$ is odd, $\tau^{w}({\bm{a}})\;(=\tau({\bm{a}}))$ equals ${\bm{a}}$ if and only if $a_{i}=-a_{i}+2\Delta$ for $i=1,\dots,n$, where $\Delta=a_{n}-a_{n-1}+\cdots+a_{1}$. This implies that $a_{i}=\Delta$ for any $i$ and hence $a_{1}=\cdots=a_{n}$; this concludes the theorem. ∎