Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs

In knot theory, minimum numbers of colors for arc colorings have been studied in many papers (see [1, 2, 4, 5, 9, 10, 11] for example). We denote by $\text{mincol}^{\text{Fox}}_{p}(K)$ the number for a Fox $p$-colorable knot $K$. As one of properties for $\text{mincol}^{\text{Fox}}_{p}(K)$, the following result is obtained in [8].

• •

  For any odd prime number $p$ and any Fox $p$-colorable knot $K$,

  $$\text{mincol}^{\text{Fox}}_{p}(K)\geq\lfloor\log_{2}p\rfloor+2.$$

Note that for the cases when $p$ is a composite number or $K$ is a link with multiple components, the similar results are known under some condition (see [6]).

On the other hand, a Dehn $p$-coloring is well-known as one of region colorings of knot diagrams which is corresponding to a Fox $p$-coloring. However, minimum numbers of colors for Dehn $p$-colorings almost have not been studied yet.

In this paper, we will consider minimum numbers of colors of Dehn $p$-colorable knots for an odd prime number $p$. The number for a Dehn $p$-colorable knot $K$ is denoted by ${\rm mincol}^{\rm Dehn}_{p}(K)$. We show in Theorem 1.5 that

• •

  for any odd prime number $p$ and any Dehn $p$-colorable knot $K$,

  $${\rm mincol}^{\rm Dehn}_{p}(K)\geq\lfloor\log_{2}p\rfloor+2.$$

Moreover, we will define the $\mathcal{R}$-palette graph for a set of colors. The $\mathcal{R}$-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn $p$-colored diagram (see Theorem 5.1). In particular, in Theorem 5.3, we give the sets of $\lfloor\log_{2}p\rfloor+2$ colors which might realize a nontrivially Dehn $p$-colored diagram for an odd prime number $p$ with $p<2^{5}$. Furthermore, we show in Proposition 5.4 that there exists a knot $K$ with ${\rm mincol}^{\rm Dehn}_{p}(K)=\lfloor\log_{2}p\rfloor+2$ for some $p$.

This paper is organized as follows: In Section 1, we review the definitions and some basic properties of Dehn colorings and minimum numbers of colors for Dehn colorings. Besides, our main result (Theorem 1.5) is stated in this section. Section 2 is devoted to setting up some extended coloring matrices which are used in Section 3. Theorem 1.5 is proven in Section 3. In Section 4, the $\mathcal{R}$-palette graph of a set of colors and that of a Dehn $p$-colored diagram are defined. In Section 5, we give candidates of sets of colors each of which might realize a nontrivially Dehn $p$-colored diagram by using the $\mathcal{R}$-palette graphs. Moreover in Appendix A, we show that ${\rm mincol}^{\rm Dehn}_{5}(K)=4$ for any Dehn $5$-colorable knot $K$.

In this paper, for a prime number $p$, we denote by $\mathbb{Z}_{p}$ the cyclic group $\mathbb{Z}/p\mathbb{Z}$. When $p=0$, read all the parts of this paper by replacing all $\mathbb{Z}_{p}$ and $p$ with $\mathbb{Z}$.

Let $p$ be an odd prime number or $p=0$. Let $D$ be a diagram of a knot $K$ and $\mathcal{R}(D)$ the set of regions of $D$. A Dehn $p$-coloring of $D$ is a map $C:\mathcal{R}(D)\to\mathbb{Z}_{p}$ satisfying the following condition:

• •

  for each crossing $c$ with regions $x_{1},x_{2},x_{3}$, and $x_{4}$ as depicted in Figure 1,

  $$C(x_{1})+C(x_{3})=C(x_{2})+C(x_{4})$$

  holds, where the region $x_{2}$ is adjacent to $x_{1}$ by an under-arc and $x_{3}$ is adjacent to $x_{1}$ by the over-arc.

We call $C(x)$ the color of a region $x$ by $C$. In this paper, as shown in the right of Figure 1, we represent a Dehn $p$-coloring $C$ of a knot diagram $D$ by assigning the color $C(x)$ to each region $x$. We mean by $(D,C)$ a diagram $D$ given a Dehn $p$-coloring $C$, and call it a Dehn $p$-colored diagram. We denote by $\mathcal{C}(D,C)$ the set of colors assigned to a region of $D$ by $C$, that is $\mathcal{C}(D,C)={\rm Im}\,C$. The set of Dehn $p$-colorings of $D$ is denoted by ${\rm Col}_{p}(D)$. We remark that the number $\#{\rm Col}_{p}(D)$ is an invariant of the knot $K$.

Let $c$ be a crossing of $D$ with regions $x_{1},x_{2},x_{3}$, and $x_{4}$ as depicted in Figure 1. We say that $c$ of $(D,C)$ is trivially colored if

hold, and nontrivially colored otherwise. A Dehn $p$-coloring $C$ of $D$ is trivial if each crossing of $(D,C)$ is trivially colored and nontrivial otherwise. We note that trivial $p$-colorings are separated into two types: monochromatic colorings and checkerboard colorings with two colors. We call the former $1$-trivial colorings and denote them by $C^{\rm 1T}$. We call the latter $2$-trivial colorings and denote them by $C^{\rm 2T}$.

A knot $K$ is Dehn $p$-colorable if $K$ has a Dehn $p$-colored diagram $(D,C)$ such that $C$ is nontrivial.

For a semiarc $u$ of a Dehn $p$-colored diagram $(D,C)$, when the two regions $x_{1}$ and $x_{2}$ which are on the both side of $u$ have $C(x_{1})=a_{1}$ and $C(x_{2})=a_{2}$ for some $a_{1},a_{2}\in\mathbb{Z}_{p}$, we call $u$ an $\{a_{1},a_{2}\}$-semiarc, where $\{a_{1},a_{2}\}$ is regarded as a multiset $\{a_{1},a_{1}\}$ when $a_{1}=a_{2}$. Moreover, we call an arc $u$ including an $\{a_{1},a_{2}\}$-semiarc an $\overline{a_{1}+a_{2}}$-arc, where we note that the value $\overline{a_{1}+a_{2}}$ does not depend on the choice of a semiarc included in $u$.

The set ${\rm Col}_{p}(D)$ can be regarded as a $\mathbb{Z}_{p}$-module with the scalar product $sC$ and the addition $C+C^{\prime}$ with

Moreover, since for any $t\in\mathbb{Z}_{p}$, the constant map $C^{\rm 1T}_{t}:\mathcal{R}(D)\to\mathbb{Z}_{p};C^{\rm 1T}_{t}(x)=t$ is a kind of Dehn $p$-coloring, the map

is also a Dehn $p$-coloring, and thus, the transformations $C\mapsto sC+t$ are closed in ${\rm Col}_{p}(D)$. In particular, a transformation $C\mapsto sC+t$ with $s\in\mathbb{Z}_{p}^{\times}$ is called a regular affine transformation on ${\rm Col}_{p}(D)$. Two Dehn $p$-colorings $C,C^{\prime}\in{\rm Col}_{p}(D)$ are affine equivalent (written $C\sim C^{\prime}$) if they are related by a regular affine transformation on ${\rm Col}_{p}(D)$, that is, $C^{\prime}=sC+t$ for some $s\in\mathbb{Z}_{p}^{\times}$ and $t\in\mathbb{Z}_{p}$.

A regular affine transformation on the $\mathbb{Z}_{p}$-module $\mathbb{Z}_{p}$ is defined by $a\mapsto sa+t$ for some $s\in\mathbb{Z}_{p}^{\times}$ and $t\in\mathbb{Z}_{p}$. Two subsets $S,S^{\prime}\subset\mathbb{Z}_{p}$ are affine equivalent (written $S\sim S^{\prime}$) if they are related by a regular affine transformation on $\mathbb{Z}_{p}$, that is, $S^{\prime}=sS+t$ for some $s\in\mathbb{Z}_{p}^{\times}$ and $t\in\mathbb{Z}_{p}$. We note that since any regular affine transformation is a bijection, $\#S=\#S^{\prime}$ holds if $S\sim S^{\prime}$.

In this paper, we focus on the minimum number of colors.

The next theorem is our main result, which will be proven in Section 3.

From now on, for an integer matrix $M$, ${\rm rank}_{p}M$ means the rank of $M$ regarded as a matrix on $\mathbb{Z}_{p}$. Note again that ${\rm rank}_{0}M$ is replaced with ${\rm rank}_{\mathbb{Z}}M$.

Let $D$ be a diagram of a knot $K$. Let $x_{1},\ldots,x_{n+2}$ be the regions, and $c_{1},\ldots,c_{n}$ the crossings of $D$. We set an arbitrary orientation for $D$.

For each crossing $c_{i}$, we set the equation

where $x_{i_{1}},x_{i_{2}},x_{i_{3}},x_{i_{4}}$ are the regions around $c_{i}$ such that $x_{i_{1}}$ is in the right side of both of over- and under-arcs, $x_{i_{2}}$ is adjacent to $x_{i_{1}}$ by an under-arc and $x_{i_{3}}$ is adjacent to $x_{i_{1}}$ by the over-arc (see the left of Figure 3), and where we regard $x_{1},\ldots,x_{n+2}$ as variables of the equation. We then have the simultaneous equations

that is

The coefficient matrix $M^{\rm col}_{D}$ is called the Dehn coloring matrix of $D$ as an unoriented diagram since the $\mathbb{Z}_{p}$-module of solutions of (2.1) for an odd prime $p$ or $p=0$ is isomorphic to the $\mathbb{Z}_{p}$-module ${\rm Col}_{p}(D)$ by the isomorphism which sends a solution ${}^{t}(a_{1}\cdots a_{n+2})$ to the Dehn coloring $C$ such that $C(x_{i})=a_{i}$ for each $i\in\{1,\ldots,n+2\}$.

It is known that the knot group $G_{K}$ of $K$, which is the fundamental group of $\mathbb{R}^{3}\setminus K$, has the presentation

where for each crossing $c_{i}$ $(1\leq i\leq n)$ as in the left of Figure 3, we set the relator $r_{i}$ by

Let $A_{D}(t)$ denote the Alexander matrix of $G_{K}$ obtained from the above presentation (2.2) by using the Fox calculus (see [3]), that is $\displaystyle A_{D}(t)=\left(\tilde{\alpha}\circ\tilde{\pi}\left(\frac{\partial r_{i}}{\partial x_{j}}\right)\right)$, where $\frac{\partial}{\partial x_{j}}$ is the Fox derivative $\mathbb{Z}F\rightarrow\mathbb{Z}F$ with $F=\langle x_{1},\ldots,x_{n+2}\rangle$, $\tilde{\pi}$ is the projection $\mathbb{Z}F\rightarrow\mathbb{Z}G_{K}$, and $\tilde{\alpha}$ is the abelianization $\mathbb{Z}G_{K}\rightarrow\mathbb{Z}\langle t\rangle=\mathbb{Z}[t,t^{-1}]$ such that a meridian element is sent to $t$. For the $i$th row vector of $A_{D}(-1)$ $(1\leq i\leq n)$, the $j$th entry is $1$ if $j=i_{1}$ or $i_{3}$, $-1$ if $j=i_{2}$ or $i_{4}$, and $0$ otherwise, which follows from

(refer to the right of Figure 3), where we here suppose that $x_{i_{1}},\ldots,x_{i_{4}}$ differ from each other. Thus we have

The greatest common divisor of all the ($n+1$)-minors of $A_{D}(-1)$ is called the determinant of $K$, which is denoted by ${\rm det}(K)$.

For an odd prime number $p$, we suppose that there exists a nontrivial Dehn $p$-coloring, say $C_{0}$, satisfying that for some $i,j\in\{1,\ldots,n+2\}$ such that $i<j$, $C_{0}(x_{i})=C_{0}(x_{j})$, $C^{\rm 2T}(x_{i})\not=C^{\rm 2T}(x_{j})$ holds. Then we set

where for the last row vector, the $i$th and $j$th entries are $1$ and $-1$, respectively, and the other entries are $0$.

Let us consider the ranks of the matrices $M_{D}^{\rm col}$, $A_{D}(-1)$ and $B_{D,C_{0};i,j}$.

For an odd prime $p$ or $p=0$, we have at least two linearly independent solutions of (2.1) on $\mathbb{Z}_{p}$: one, denoted by $\boldsymbol{a}^{\rm 1T}$, is corresponding to the 1-trivial Dehn $\mathbb{Z}_{p}$-coloring $C^{\rm 1T}$ with $C^{\rm 1T}(x_{1})=1$, and the other, denoted by $\boldsymbol{a}^{\rm 2T}$, is corresponding to a 2-trivial Dehn $\mathbb{Z}_{p}$-coloring $C^{\rm 2T}$ with $C^{\rm 2T}(x_{1})=0$ and ${\rm Im}C^{\rm 2T}=\{0,1\}$. Thus we can see that ${\rm rank}_{p}M^{\rm col}_{D}\leq n$.

Besides, it is well-known that ${\rm det}(K)\not=0$, which gives ${\rm rank}_{\mathbb{Z}}A_{D}(-1)=n+1$, and hence, ${\rm rank}_{\mathbb{Z}}M_{D}^{\rm col}\geq n$. Considering this property together with the fact that ${\rm rank}_{\mathbb{Z}}M_{D}^{\rm col}\leq n$, we have ${\rm rank}_{\mathbb{Z}}M_{D}^{\rm col}=n$, and indeed, the $\mathbb{Z}$-module of solutions of (2.1) on $\mathbb{Z}$ is spanned by two vectors $\boldsymbol{a}^{\rm 1T}$ and $\boldsymbol{a}^{\rm 2T}$.

On the other hand, for an odd prime $p$, if $D$ admits a nontrivial Dehn $p$-coloring $C$, then there exists a solution $\boldsymbol{a}$ of (2.1) on $\mathbb{Z}_{p}$ such that $\boldsymbol{a}^{\rm 1T}$, $\boldsymbol{a}^{\rm 2T}$ and $\boldsymbol{a}$ are linearly independent. Hence, for this case, ${\rm rank}_{p}M^{\rm col}_{D}\leq n-1$.

For the matrix $A_{D}(-1)$, ${\rm rank}_{\mathbb{Z}}A_{D}(-1)=n+1$ as mentioned above. For an odd prime $p$ or $p=0$, note that the $\mathbb{Z}_{p}$-module of solutions of the simultaneous equations

on $\mathbb{Z}_{p}$ is a submodule of that of (2.1) on $\mathbb{Z}_{p}$. Any vector $k\boldsymbol{a}^{\rm 1T}+\ell\boldsymbol{a}^{\rm 2T}$ ($k,\ell\in\mathbb{Z}$, $k\not=0$) can not be a solution of (2.3) on $\mathbb{Z}_{p}$ since any solution $\boldsymbol{a}=\,^{t}(a_{1}\cdots a_{n+2})$ of (2.3) needs to satisfy $a_{1}=0$, while $\boldsymbol{a}^{\rm 2T}$ is a linearly independent solution of (2.3) on $\mathbb{Z}_{p}$. This implies that the $\mathbb{Z}$-module of solutions of (2.3) on $\mathbb{Z}$ is spanned by $\boldsymbol{a}^{\rm 2T}$.

On the other hand, for an odd prime $p$, if $D$ admits a nontrivial Dehn $p$-coloring $C$ such that $C(x_{1})=0$, then there exists a solution $\boldsymbol{a}$ of (2.3) on $\mathbb{Z}_{p}$ such that $\boldsymbol{a}^{\rm 2T}$ and $\boldsymbol{a}$ are linearly independent. Hence, for this case, ${\rm rank}_{p}A_{D}(-1)\leq n$.

For the matrix $B_{D,C_{0};i,j}$, we note that it is defined for a given odd prime $p$, a Dehn $p$-coloring $C_{0}$, and $i,j\in\{1,\ldots,n+2\}$. Note that the $\mathbb{Z}$- (resp. $\mathbb{Z}_{p}$-)module of solutions of the simultaneous equations

on $\mathbb{Z}$ (resp. $\mathbb{Z}_{p}$) is a submodule of that of (2.1) on $\mathbb{Z}$ (resp. $\mathbb{Z}_{p}$).

Any vector $k\boldsymbol{a}^{\rm 1T}+\ell\boldsymbol{a}^{\rm 2T}$ ($k,\ell\in\mathbb{Z}$, $\ell\not=0$) can not be a solution of (2.4) on $\mathbb{Z}$ (resp. $\mathbb{Z}_{p}$) since any solution $\boldsymbol{a}=\,^{t}(a_{1}\cdots a_{n+2})$ of (2.4) needs to satisfy $a_{i}=a_{j}$, while $\boldsymbol{a}^{\rm 1T}$ is a linearly independent solution of (2.4) on $\mathbb{Z}$ (resp. $\mathbb{Z}_{p}$). Hence the $\mathbb{Z}$-module of solutions of (2.4) on $\mathbb{Z}$ is spanned by $\boldsymbol{a}^{\rm 1T}$, which implies that ${\rm rank}_{\mathbb{Z}}B_{D,C_{0};i,j}=n+1$.

On the other hand, since $C_{0}$ is a nontrivial $p$-coloring, there exists a solution $\boldsymbol{a}_{0}=\,^{t}(C_{0}(x_{1})\cdots C_{0}(x_{n+2}))$ of (2.4) on $\mathbb{Z}_{p}$, and $\boldsymbol{a}^{\rm 1T}$ and $\boldsymbol{a}_{0}$ are linearly independent. Hence, ${\rm rank}_{p}B_{D,C_{0};i,j}\leq n$.

As a summary, we have the following properties:

Let $p$ be an odd prime integer. Let $D$ be a diagram of a knot $K$. Let $x_{1},\ldots,x_{n+2}$ be the regions, and $c_{1},\ldots,c_{n}$ the crossings of $D$. Let $C^{\rm 1T}$ and $C^{\rm 2T}$ be the $1$- and $2$-trivial Dehn $p$-coloring, respectively, such that $C^{\rm 1T}(x_{1})=1$, $C^{\rm 2T}(x_{1})=0$ and ${\rm Im}C^{\rm 2T}=\{0,1\}$. Let $C$ be a nontrivial Dehn $p$-coloring of $D$.