The Gordian distance of handlebody-knots and Alexander biquandle colorings

TL;DR

This paper introduces a method using Alexander biquandle colorings to establish lower bounds on the Gordian distance and unknotting number of handlebody-knots, and constructs examples with arbitrary distances.

Contribution

It provides a novel approach to estimate handlebody-knot complexity and constructs handlebody-knots with specified Gordian distances and unknotting numbers.

Findings

Lower bounds for Gordian distance and unknotting number established

Construction of handlebody-knots with arbitrary Gordian distance and unknotting number

Alexander biquandle colorings are effective for analyzing handlebody-knots

Abstract

We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.

Figures

Equations

$$(x\underline{\ast }y)\underline{\ast }(z\underline{\ast }y)=(x\underline{\ast }z)\underline{\ast }(y\overline{\ast }z),$$

$$(x\underline{\ast }y)\overline{\ast }(z\underline{\ast }y)=(x\overline{\ast }z)\underline{\ast }(y\overline{\ast }z),$$

$$(x\overline{\ast }y)\overline{\ast }(z\overline{\ast }y)=(x\overline{\ast }z)\overline{\ast }(y\underline{\ast }z).$$

$$a{\underline{\ast }}^{[0]}b=a,a{\underline{\ast }}^{[1]}b=a\underline{\ast }b,a{\underline{\ast }}^{[i+j]}b=(a{\underline{\ast }}^{[i]}b){\underline{\ast }}^{[j]}(b{\underline{\ast }}^{[i]}b),$$

$$a{\overline{\ast }}^{[0]}b=a,a{\overline{\ast }}^{[1]}b=a\overline{\ast }b,a{\overline{\ast }}^{[i+j]}b=(a{\overline{\ast }}^{[i]}b){\overline{\ast }}^{[j]}(b{\overline{\ast }}^{[i]}b)$$

$$\mathrm{type}X=\min \{n>0\mid a{\underline{\ast }}^{[n]}b=a=a{\overline{\ast }}^{[n]}b(\forall a,b\in X)\}.$$

$$x{\underline{\ast }}^{g}x=x{\overline{\ast }}^{g}x.$$

$$x{\underline{\ast }}^{gh}y=(x{\underline{\ast }}^{g}y){\underline{\ast }}^h(y{\underline{\ast }}^{g}y),x{\underline{\ast }}^{e}y=x,$$

$$x{\overline{\ast }}^{gh}y=(x{\overline{\ast }}^{g}y){\overline{\ast }}^h(y{\overline{\ast }}^{g}y),x{\overline{\ast }}^{e}y=x.$$

$$(x{\underline{\ast }}^{g}y){\underline{\ast }}^h(z{\overline{\ast }}^{g}y)=(x{\underline{\ast }}^{h}z){\underline{\ast }}^{h^{-1}gh}(y{\underline{\ast }}^{h}z),$$

$$(x{\overline{\ast }}^{g}y){\underline{\ast }}^h(z{\overline{\ast }}^{g}y)=(x{\underline{\ast }}^{h}z){\overline{\ast }}^{h^{-1}gh}(y{\underline{\ast }}^{h}z),$$

$$(x{\overline{\ast }}^{g}y){\overline{\ast }}^h(z{\overline{\ast }}^{g}y)=(x{\overline{\ast }}^{h}z){\overline{\ast }}^{h^{-1}gh}(y{\underline{\ast }}^{h}z).$$

$$\delta (y,y^{\prime }):=\{\begin{array}{ll}1& (y=y^{\prime }),\\ 0& (y\ne y^{\prime }).\end{array}$$

$$a_{i,j}=\{\begin{array}{ll}\delta (u_i,x_j)t^{{\psi }_i}+\delta (v_i,x_j)(s^{{\psi }_i}-t^{{\psi }_i})-\delta (w_i,x_j)& (1\le i\le n),\\ -\delta (v_{i-n},x_j)s^{{\varphi }_{i-n}}+\delta (v_{i-n}^{\prime },x_j)& (n+1\le i\le 2n),\\ \delta ({\alpha }_{i-2n},x_j)-\delta ({\gamma }_{i-2n},x_j)& (2n+1\le i\le 2n+2k),\\ \delta ({\beta }_{i-2n-2k},x_j)-\delta ({\gamma }_{i-2n-2k},x_j)s^{{\eta }_{i-2n-2k}}& (2n+2k+1\le i\le 2n+4k).\end{array}$$

$${\mathrm{Col}}_X(D,\varphi )=\left\{\left(\begin{array}{c}{z}_1\\ z_2\\ ⋮\\ z_{2n+3k}\end{array}\right)\in X^{2n+3k}|A(D,\varphi ;X)\left(\begin{array}{c}{z}_1\\ z_2\\ ⋮\\ z_{2n+3k}\end{array}\right)=0\right\}.$$

$$A(E,\psi ;X)=\left(\begin{array}{ccccccc}-1& 0& s^a-t^a& t^a& 0& 0& 0\\ 0& -1& 0& s^b-t^b& 0& t^b& 0\\ 0& 1& -s^b& 0& 0& 0& 0\\ 0& 0& 0& -s^a& 0& 0& 1\\ 0& 0& 1& 0& -1& 0& 0\\ 0& 0& 0& 0& -1& 1& 0\\ 1& 0& 0& 0& -s^a& 0& 0\\ 0& 0& 0& 0& -s^a& 0& 1\end{array}\right).$$

$$A(D,\varphi ;X)=(a_{i,j})=\left(\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ ⋮\\ {\mathbf{a}}_{2n+4k}\end{array}\right).$$

$$\sum _{i=1}^n{ϵ}_{c_i}{t}^{-\rho (w_i)}(s^{{\varphi }_i}-t^{{\varphi }_i}){\mathbf{a}}_i+\sum _{i=1}^n{ϵ}_{c_i}{t}^{-\rho (v_i^{\prime })}(s^{{\psi }_i}-t^{{\psi }_i}){\mathbf{a}}_{n+i}$$

$$+\sum _{i=1}^{2k}{ϵ}_{{\tau }_i}{t}^{-\rho ({\alpha }_i)}(s^{{\eta }_i}-t^{{\eta }_i}){\mathbf{a}}_{2n+i}+\sum _{i=1}^{2k}{ϵ}_{{\tau }_i}{t}^{-\rho ({\beta }_i)}(s^{{\theta }_i}-t^{{\theta }_i}){\mathbf{a}}_{2n+2k+i}=0.$$

$$ϵ(y;\sigma ):=\{\begin{array}{ll}1& \text{if the orientation of y points to σ},\\ -1& \text{otherwise}.\end{array}$$

$$\sum _{i=1}^n{ϵ}_{c_i}{t}^{-\rho (w_i)}(s^{{\varphi }_i}-t^{{\varphi }_i})a_{i,j}+\sum _{i=1}^n{ϵ}_{c_i}{t}^{-\rho (v_i^{\prime })}(s^{{\psi }_i}-t^{{\psi }_i})a_{n+i,j}$$

$$+\sum _{i=1}^{2k}{ϵ}_{{\tau }_i}{t}^{-\rho ({\alpha }_i)}(s^{{\eta }_i}-t^{{\eta }_i})a_{2n+i,j}+\sum _{i=1}^{2k}{ϵ}_{{\tau }_i}{t}^{-\rho ({\beta }_i)}(s^{{\theta }_i}-t^{{\theta }_i})a_{2n+2k+i,j}=0.$$

$${ϵ}_{c_i}{t}^{-\rho (w_i)}(s^{{\varphi }_i}-t^{{\varphi }_i})\delta (u_i,x_j)t^{{\psi }_i}=\delta (u_i,x_j)ϵ(u_i;c_i)t^{-\rho (u_i)}(s^{\varphi (u_i)}-t^{\varphi (u_i)}),$$

$${ϵ}_{c_i}{t}^{-\rho (w_i)}(s^{{\varphi }_i}-t^{{\varphi }_i})\delta (v_i,x_j)(s^{{\psi }_i}-t^{{\psi }_i})$$

$$={ϵ}_{c_i}{t}^{-\rho (w_i)}{s}^{{\varphi }_i}\delta (v_i,x_j)(s^{{\psi }_i}-t^{{\psi }_i})-{ϵ}_{c_i}{t}^{-\rho (w_i)}{t}^{{\varphi }_i}\delta (v_i,x_j)(s^{{\psi }_i}-t^{{\psi }_i})$$

$$={ϵ}_{c_i}{t}^{-\rho (w_i)}\delta (v_i,x_j)(s^{{\psi }_i}-t^{{\psi }_i})s^{{\varphi }_i}+\delta (v_i,x_j)ϵ(v_i;c_i)t^{-\rho (v_i)}(s^{\varphi (v_i)}-t^{\varphi (v_i)}),$$

$${ϵ}_{c_i}{t}^{-\rho (w_i)}(s^{{\varphi }_i}-t^{{\varphi }_i})(-\delta (w_i,x_j))=\delta (w_i,x_j)ϵ(w_i;c_i)t^{-\rho (w_i)}(s^{\varphi (w_i)}-t^{\varphi (w_i)}).$$

$${ϵ}_{c_i}{t}^{-\rho (v_i^{\prime })}(s^{{\psi }_i}-t^{{\psi }_i})(-\delta (v_i,x_j)s^{{\varphi }_i})=-{ϵ}_{c_i}{t}^{-\rho (v_i^{\prime })}\delta (v_i,x_j)(s^{{\psi }_i}-t^{{\psi }_i})s^{{\varphi }_i},$$

$${ϵ}_{c_i}{t}^{-\rho (v_i^{\prime })}(s^{{\psi }_i}-t^{{\psi }_i})\delta (v_i^{\prime },x_j)=\delta (v_i^{\prime },x_j)ϵ(v_i^{\prime };c_i)t^{-\rho (v_i^{\prime })}(s^{\varphi (v_i^{\prime })}-t^{\varphi (v_i^{\prime })}).$$

$${ϵ}_{{\tau }_i}{t}^{-\rho ({\alpha }_i)}(s^{{\eta }_i}-t^{{\eta }_i})\delta ({\alpha }_i,x_j)=\delta ({\alpha }_i,x_j)ϵ({\alpha }_i;{\tau }_i)t^{-\rho ({\alpha }_i)}(s^{\varphi ({\alpha }_i)}-t^{\varphi ({\alpha }_i)}),$$

$${ϵ}_{{\tau }_i}{t}^{-\rho ({\alpha }_i)}(s^{{\eta }_i}-t^{{\eta }_i})(-\delta ({\gamma }_i,x_j))=\delta ({\gamma }_i,x_j)ϵ({\gamma }_i;{\tau }_i)t^{-\rho ({\gamma }_i)}{t}^{{\theta }_i}(s^{{\eta }_i}-t^{{\eta }_i}).$$

$${ϵ}_{{\tau }_i}{t}^{-\rho ({\beta }_i)}(s^{{\theta }_i}-t^{{\theta }_i})\delta ({\beta }_i,x_j)=\delta ({\beta }_i,x_j)ϵ({\beta }_i;{\tau }_i)t^{-\rho ({\beta }_i)}(s^{\varphi ({\beta }_i)}-t^{\varphi ({\beta }_i)}),$$

$${ϵ}_{{\tau }_i}{t}^{-\rho ({\beta }_i)}(s^{{\theta }_i}-t^{{\theta }_i})(-\delta ({\gamma }_i,x_j)s^{{\eta }_i})=\delta ({\gamma }_i,x_j)ϵ({\gamma }_i;{\tau }_i)t^{-\rho ({\gamma }_i)}(s^{{\theta }_i}-t^{{\theta }_i})s^{{\eta }_i}.$$

$$(1)+(2)=\delta (v_i,x_j)ϵ(v_i;c_i)t^{-\rho (v_i)}(s^{\varphi (v_i)}-t^{\varphi (v_i)}),$$

$$(3)+(4)=\delta ({\gamma }_i,x_j)ϵ({\gamma }_i;{\tau }_i)t^{-\rho ({\gamma }_i)}(s^{\varphi ({\gamma }_i)}-t^{\varphi ({\gamma }_i)}).$$

$$\sum _{i=1}^n{ϵ}_{c_i}{t}^{-\rho (w_i)}(s^{{\varphi }_i}-t^{{\varphi }_i})a_{i,j}+\sum _{i=1}^n{ϵ}_{c_i}{t}^{-\rho (v_i^{\prime })}(s^{{\psi }_i}-t^{{\psi }_i})a_{n+i,j}$$

$$+\sum _{i=1}^{2k}{ϵ}_{{\tau }_i}{t}^{-\rho ({\alpha }_i)}(s^{{\eta }_i}-t^{{\eta }_i})a_{2n+i,j}+\sum _{i=1}^{2k}{ϵ}_{{\tau }_i}{t}^{-\rho ({\beta }_i)}(s^{{\theta }_i}-t^{{\theta }_i})a_{2n+2k+i,j}$$

$$=\sum _{i=1}^n(\delta (u_i,x_j)ϵ(u_i;c_i)t^{-\rho (u_i)}(s^{\varphi (u_i)}-t^{\varphi (u_i)})$$

$$+\delta (v_i,x_j)ϵ(v_i;c_i)t^{-\rho (v_i)}(s^{\varphi (v_i)}-t^{\varphi (v_i)})$$

$$+\delta (v_i^{\prime },x_j)ϵ(v_i^{\prime };c_i)t^{-\rho (v_i^{\prime })}(s^{\varphi (v_i^{\prime })}-t^{\varphi (v_i^{\prime })})$$

Full text

1. Introduction

The Gordian distance of two classical knots is the minimal number of crossing changes neened to be deformed each other. In particular, we call the Gordian distance of a classical knot and the trivial one the unknotting number of the classical knot. Clark, Elhamdadi, Saito and Yeatman [2] gave a lower bound for the Nakanishi index [18], which induced a lower bound for the unknotting number of classical knots. This is an generalization of the Przytycki’s result [19]. In this paper, we give lower bounds for the Gordian distance and the unknotting number of handlebody-knots, which is a generalization of a classical knot with respect to a genus.

Ishii [5] introduced an enhanced constituent link of a spatial trivalent graph, and Ishii and Iwakiri [8] introduced an $A$-flow of a spatial graph, where $A$ is an abelian group, to define colorings and invariants of handlebody-knots. Iwakiri [14] gave a lower bound for the unknotting number of handlebody-knots by using Alexander quandle colorings of its $\mathbb{Z}_{2}$ or $\mathbb{Z}_{3}$-flowed diagram. Ishii, Iwakiri, Jang and Oshiro [9] introduced a $G$-family of quandles, which is an extension of the above structures. Recently, Ishii and Nelson [13] introduced a $G$-family of biquandles, which is a biquandle version of a $G$-family of quandles.

In this paper, we extend the result in [14] in three directions. First, we extend from $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$-flows to any $\mathbb{Z}_{m}$-flow. Second, we extend from quandles to biquandles. Finally, we extend from unknotting numbers to Gordian distances. Thus we can determine the Gordian distance and the unknotting number of handlebody-knots more efficiently. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any $n\in\mathbb{Z}_{>0}$ and note that one of them can not be obtained by using Alexander quandle colorings introduced in [14].

This paper is organized into seven sections. In Section 2, we recall the definition of a handlebody-knot and introduce the Gordian distance and the unknotting number of handlebody-knots. In Section 3, we recall the definition of a (bi)quandle and a $G$-family of (bi)quandles. In Section 4, we introduce a coloring of a diagram of a handlebody-knot by using a $G$-family of biquandles. In Section 5, we show that there are linear relationships for Alexander biquandle colorings of a diagram of a handlebody-knot. In Section 6, we give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using $\mathbb{Z}_{m}$-family of Alexander biquandles colorings. In section 7, we construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any $n\in\mathbb{Z}_{>0}$. Moreover, we note that one of them can not be obtained by using Alexander quandle colorings with $\mathbb{Z}_{2},\mathbb{Z}_{3}$-flows introduced in [14].

2. The Gordian distance of handlebody-knots

A handlebody-link, which is introduced in [5], is the disjoint union of handlebodies embedded in the 3-sphere $S^{3}$. A handlebody-knot is a handlebody-link with one component. In this paper, we assume that every component of a handlebody-link is of genus at least $1$. An $S^{1}$-orientation of a handlebody-link is an orientation of all genus 1 components of the handlebody-link, where an orientation of a solid torus is an orientation of its core $S^{1}$. Two $S^{1}$-oriented handlebody-links are equivalent if there exists an orientation-preserving self-homeomorphism of $S^{3}$ sending one to the other preserving the $S^{1}$-orientation.

A spatial trivalent graph is a graph whose vertices are valency 3 embedded in $S^{3}$. In this paper, a trivalent graph may have a circle component, which has no vertices. A Y-orientation of a spatial trivalent graph is a direction of all edges of the graph satisfying that every vertex of the graph is both the initial vertex of a directed edge and the terminal vertex of a directed edge (Figure 1). A vertex of a Y-oriented spatial trivalent graph can be allocated a sign; the vertex is said to be positive or negative, or to have sign $+1$ or $-1$. The standard convention is shown in Figure 1. For a Y-oriented spatial trivalent graph $K$ and an $S^{1}$-oriented handlebody-link $H$, we say that $K$ represents $H$ if $H$ is a regular neighborhood of $K$ and the $S^{1}$-orientation of $H$ agrees with the Y-orientation. Then any $S^{1}$-oriented handlebody-link can be represented by some Y-oriented spatial trivalent graph. We define a diagram of an $S^{1}$-oriented handlebody-link by a diagram of a Y-oriented spatial trivalent graph representing the handlebody-link. An $S^{1}$-oriented handlebody-link is trivial if it has a diagram with no crossings. Then the following theorem holds.

Theorem 2.1** ([7]).**

For a diagram $D_{i}$ of an $S^{1}$-oriented handlebody-link $H_{i}$ $(i=1,2)$, $H_{1}$ and $H_{2}$ are equivalent if and only if $D_{1}$ and $D_{2}$ are related by a finite sequence of R1–R6 moves depicted in Figure 2 preserving Y-orientations.

In this paper, for a diagram $D$ of an $S^{1}$-oriented handlebody-link, we denote by $\mathcal{A}(D)$ and $\mathcal{SA}(D)$ the set of all arcs of $D$ and the one of all semi-arcs of $D$ respectively, where a semi-arc is a piece of a curve each of whose endpoints is a crossing or a vertex. An orientation of a (semi-)arc of $D$ is also represented by the normal orientation obtained by rotating the usual orientation counterclockwise by $\pi/2$ on the diagram. For any $m\in\mathbb{Z}_{\geq 0}$, we put $\mathbb{Z}_{m}:=\mathbb{Z}/m\mathbb{Z}$.

A crossing change of an $S^{1}$-oriented handlebody-link $H$ is that of a spatial trivalent graph representing $H$. This deformation can be realized by switching two handles depicted in Figure 3. It is easy to see that any two $S^{1}$-oriented handlebody-knots of the same genus can be related by a finite sequence of crossing changes. For any two $S^{1}$-oriented handlebody-knots $H_{1}$ and $H_{2}$ of the same genus, we define their Gordian distance $d(H_{1},H_{2})$ by the minimal number of crossing changes neened to be deformed each other. In particular, for any $S^{1}$-oriented handlebody-knot $H$ and the $S^{1}$-oriented trivial handlebody-knot $O$ of the same genus, we define $u(H):=d(H,O)$, which is called the unknotting number of $H$.

3. A biquandle and a $G$-family of biquandles

We recall the definitions of a quandle and a biquandle.

Definition 3.1** ([15, 16]).**

A quandle is a non-empty set $X$ with a binary operation $*:X\times X\to X$ satisfying the following axioms.

• •

For any $x\in X$, $x*x=x$.

• •

For any $x\in X$, the map $S_{x}:X\to X$ defined by $S_{x}(y)=y*x$ is a bijection.

• •

For any $x,y,z\in X$, $(x*y)*z=(x*z)*(y*z)$.

Definition 3.2** ([3]).**

A biquandle is a non-empty set $X$ with binary operations $\mathbin{\overline{*}},\mathbin{\underline{*}}:X\times X\to X$ satisfying the following axioms.

• •

For any $x\in X$, $x\mathbin{\underline{*}}x=x\mathbin{\overline{*}}x$.

• •

For any $x\in X$, the map $\underline{S}_{x}:X\to X$ defined by $\underline{S}_{x}(y)=y\mathbin{\underline{*}}x$ is a bijection.

• For any $x\in X$, the map $\overline{S}_{x}:X\to X$ defined by $\overline{S}_{x}(y)=y\mathbin{\overline{*}}x$ is a bijection.
• The map $S:X\times X\to X\times X$ defined by $S(x,y)=(y\mathbin{\overline{*}}x,x\mathbin{\underline{*}}y)$ is a bijection.
• •

For any $x,y,z\in X$,

[TABLE]

We define $\mathbin{\underline{*}}^{n}x:=\underline{S}_{x}^{n}$ and $\mathbin{\overline{*}}^{n}x:=\overline{S}_{x}^{n}$ for any $n\in\mathbb{Z}$. We note that $(X,*)$ is a quandle if and only if $(X,*,\mathbin{\overline{*}})$ is a biquandle with $x\mathbin{\overline{*}}y=x$. For any $m\in\mathbb{Z}_{\geq 0}$, a $\mathbb{Z}_{m}[s^{\pm 1},t^{\pm 1}]$-module $X$ is a biquandle with $a\mathbin{\underline{*}}b=ta+(s-t)b$ and $a\mathbin{\overline{*}}b=sa$, which we call an Alexander biquandle. When $s=1$, an Alexander biquandle coincides with an Alexander quandle.

Definition 3.3** ([10]).**

Let $X$ be a biquandle. We define two families of binary operations $\mathbin{\underline{*}}^{[n]},\mathbin{\overline{*}}^{[n]}:X\times X\to X(n\in\mathbb{Z})$ by the equalities

[TABLE]

for any $i,j\in\mathbb{Z}$.

Since $a=a\mathbin{\underline{*}}^{[0]}b=(a\mathbin{\underline{*}}^{[-1]}b)\mathbin{\underline{*}}^{[1]}(b\mathbin{\underline{*}}^{[-1]}b)=(a\mathbin{\underline{*}}^{[-1]}b)\mathbin{\underline{*}}(b\mathbin{\underline{*}}^{[-1]}b)$, we have $a\mathbin{\underline{*}}^{[-1]}b=a\mathbin{\underline{*}}^{-1}(b\mathbin{\underline{*}}^{[-1]}b)$ and $(b\mathbin{\underline{*}}^{[-1]}b)\mathbin{\underline{*}}(b\mathbin{\underline{*}}^{[-1]}b)=b$. Then for an Alexander biquandle $X$, we have $a\mathbin{\underline{*}}^{[n]}b=t^{n}a+(s^{n}-t^{n})b$ and $a\mathbin{\overline{*}}^{[n]}b=s^{n}a$ for any $a,b\in X$.

We define the type of a biquandle $X$ by

[TABLE]

Any finite biquandle is of finite type [13].

We also recall the definitions of a $G$-family of quandles and a $G$-family of biquandles.

Definition 3.4** ([9]).**

Let $G$ be a group with the identity element $e$. A $G$-family of quandles is a non-empty set $X$ with a family of binary operations $*^{g}:X\times X\to X~{}(g\in G)$ satisfying the following axioms.

• •

For any $x\in X$ and $g\in G$, $x*^{g}x=x.$

• •

For any $x,y\in X$ and $g,h\in G$, $x*^{gh}y=(x*^{g}y)*^{h}y$ and $x*^{e}y=x$.

• •

For any $x,y,z\in X$ and $g,h\in G$, $(x*^{g}y)*^{h}z=(x*^{h}z)*^{h^{-1}gh}(y*^{h}z)$.

Definition 3.5** ([10, 13]).**

Let $G$ be a group with the identity element $e$. A $G$-family of biquandles is a non-empty set $X$ with two families of binary operations $\mathbin{\underline{*}}^{g},\mathbin{\overline{*}}^{g}:X\times X\to X~{}(g\in G)$ satisfying the following axioms.

• •

For any $x\in X$ and $g\in G$,

[TABLE]

• •

For any $x,y\in X$ and $g,h\in G$,

[TABLE]

• •

For any $x,y,z\in X$ and $g,h\in G$,

[TABLE]

For a biquandle $(X,\mathbin{\underline{*}},\mathbin{\overline{*}})$ with $\operatorname{type}X<\infty$, $(X,\{\mathbin{\underline{*}}^{[n]}\}_{[n]\in\mathbb{Z}_{\operatorname{type}X}},\{\mathbin{\overline{*}}^{[n]}\}_{[n]\in\mathbb{Z}_{\operatorname{type}X}})$ is a $\mathbb{Z}_{\operatorname{type}X}$-family of biquandles [13]. In particular, when $X$ is an Alexander biquandle, $(X,\{\mathbin{\underline{*}}^{[n]}\}_{[n]\in\mathbb{Z}_{\operatorname{type}X}},\{\mathbin{\overline{*}}^{[n]}\}_{[n]\in\mathbb{Z}_{\operatorname{type}X}})$ is called a $\mathbb{Z}_{\operatorname{type}X}$-family of Alexander biquandles.

4. Colorings

In this section, we introduce a coloring of a diagram of an $S^{1}$-oriented handlebody-link by a $G$-family of biquandles. Let $G$ be a group and let $D$ be a diagram of an $S^{1}$-oriented handlebody-link $H$. A $G$-flow of $D$ is a map $\phi:\mathcal{A}(D)\to G$ satisfying

at each crossing and each vertex. In this paper, to avoid confusion, we often represent an element of $G$ with an underline. We denote by $(D,\phi)$, which is called a $G$-flowed diagram of $H$, a diagram $D$ given a $G$-flow $\phi$ and put $\mathrm{Flow}(D;G):=\{\phi\mid\phi:G\textrm{-flow~{}of~{}}D\}$. We can identify a $G$-flow $\phi$ with a homomorphism from the fundamental group $\pi_{1}(S^{3}-H)$ to $G$.

Let $G$ be a group and let $D$ be a diagram of an $S^{1}$-oriented handlebody-link $H$. Let $D^{\prime}$ be a diagram obtained by applying one of Reidemeister moves to the diagram $D$ once. For any $G$-flow $\phi$ of $D$, there is an unique $G$-flow $\phi^{\prime}$ of $D^{\prime}$ which coincides with $\phi$ except near the point where the move applied. Therefore the number of $G$-flow of $D$, denoted by $\#\mathrm{Flow}(D;G)$, is an invariant of $H$. We call the $G$-flow $\phi^{\prime}$ the associated $G$-flow of $\phi$ and the $G$-flowed diagram $(D^{\prime},\phi^{\prime})$ the associated $G$-flowed diagram of $(D,\phi)$.

For any $m\in\mathbb{Z}_{\geq 0}$ and $\mathbb{Z}_{m}$-flow $\phi$ of a diagram $D$ of an $S^{1}$-oriented handlebody-link $H$, we define $\gcd\phi:=\gcd\{\phi(a),m\mid a\in\mathcal{A}(D)\}$. Then we have the following lemma in the same way as in [11].

Lemma 4.1**.**

For any $m\in\mathbb{Z}_{\geq 0}$, let $(D,\phi)$ be a $\mathbb{Z}_{m}$-flowed diagram of an $S^{1}$-oriented handlebody-link H and let $(D^{\prime},\phi^{\prime})$ be the associated $\mathbb{Z}_{m}$-flowed diagram of $(D,\phi)$. Then it follows that $\gcd\phi=\gcd\phi^{\prime}$.

Let $G$ be a group, $X$ be a $G$-family of biquandles and let $(D,\phi)$ be a $G$-flowed diagram of an $S^{1}$-oriented handlebody-link $H$. An $X$-coloring of $(D,\phi)$ is a map $C:\mathcal{SA}(D,\phi)\to X$ satisfying

at each crossing and each vertex, where $\mathcal{SA}(D,\phi)$ is the set of all semi-arcs of $(D,\phi)$. We denote by $\mathrm{Col}_{X}(D,\phi)$ the set of all $X$-colorings of $(D,\phi)$. We note that $\mathrm{Col}_{X}(D,\phi)$ is a vector space over $X$ when $X$ is a field.

Proposition 4.2** ([13]).**

Let $X$ be a $G$-family of biquandles and let $(D,\phi)$ be a $G$-flowed diagram of an $S^{1}$-oriented handlebody-link H. Let $(D^{\prime},\phi^{\prime})$ be the associated $G$-flowed diagram of $(D,\phi)$. For any $X$-coloring $C$ of $(D,\phi)$, there is an unique $X$-coloring $C^{\prime}$ of $(D^{\prime},\phi^{\prime})$ which coincides with $C$ except near the point where the move applied.

We call the $X$-coloring $C^{\prime}$ the associated $X$-coloring of $C$. By this proposition, we have $\#\mathrm{Col}_{X}(D,\phi)=\#\mathrm{Col}_{X}(D^{\prime},\phi^{\prime})$.

Proposition 4.3**.**

Let $G$ be a group and let X be a $G$-family of biquandles. Then the following hold.

1. (1)

Let $(D,\phi)$ be a $G$-flowed diagram of an $S^{1}$-oriented handlebody-link. Then it follows that $\#\mathrm{Col}_{X}(D,\phi)\geq\#X$. 2. (2)

Let $(O,\psi)$ be a $G$-flowed diagram of an $S^{1}$-oriented $m$-component trivial handlebody-link. Then it follows that $\#\mathrm{Col}_{X}(O,\psi)=(\#X)^{m}$.

Proof.

1. (1)

By Theorem 2.1 and [17], we can deform $(D,\phi)$ into the $G$-flowed diagram $(D^{\prime},\phi^{\prime})$ depicted in Figure 4 by a finite sequence of Reidemeinter moves preserving Y-orientations, where $b$ is a classical $l$-braid, and $a_{i,1},\ldots,a_{i,m_{i}},b_{i,1},\ldots,b_{i,n_{i}}\in G$ for any $i=1,\ldots,s$. We note that $\prod_{j=1}^{m_{i}}a_{i,j}=\prod_{j=1}^{n_{i}}b_{i,j}$ for any $i=1,\ldots,s$, and $x\mathbin{\underline{*}}^{g}x=x\mathbin{\overline{*}}^{g}x$ for any $x\in X$ and $g\in G$. By Proposition 4.2, it is sufficient to prove that $\#\mathrm{Col}_{X}(D^{\prime},\phi^{\prime})\geq\#X$. Here for any $x\in X$ and $g\in G$, we write $x*^{g}x$ for $x\mathbin{\underline{*}}^{g}x$ and $x\mathbin{\overline{*}}^{g}x$ simply. Then for any $x\in X$, the assignment of elements of $X$ to each semi-arc of $(D^{\prime},\phi^{\prime})$ as shown in Figures 4 and 5 is an $X$-coloring, where each $g_{i}$ represents an element of $G$ in Figure 5. Therefore we have $\#\mathrm{Col}_{X}(D^{\prime},\phi^{\prime})\geq\#X$.

2. (2)

It is sufficient to prove that $\#\mathrm{Col}_{X}(O,\psi)=\#X$ when $m=1$. Let $(O_{g},\psi_{g})$ be a $G$-flowed diagram of an $S^{1}$-oriented trivial handlebody-knot of genus $g$. By Theorem 2.1, we can deform $(O_{g},\psi_{g})$ into the $G$-flowed diagram $(O^{\prime}_{g},\psi^{\prime}_{g})$ depicted in Figure 6 by a finite sequence of Reidemeinter moves preserving Y-orientations, where $a_{i}\in G$ for any $i=1,\ldots,g$, and $e$ is the identity of $G$. By Proposition 4.2, it is sufficient to prove that $\#\mathrm{Col}_{X}(O^{\prime}_{g},\psi^{\prime}_{g})=\#X$. For any $x\in X$, the assignment of $x$ to each semi-arc of $(O^{\prime}_{g},\psi^{\prime}_{g})$ as shown in Figure 6 is an $X$-coloring. On the other hand, since any $X$-coloring of $(O_{g}^{\prime},\psi^{\prime}_{g})$ is given by Figure 6 for some $x\in X$, we have $\#\mathrm{Col}_{X}(O_{g}^{\prime},\psi^{\prime}_{g})=\#X$.

∎

5. Linear relationships for Alexander biquandle colorings

For any $\mathbb{Z}_{m}$-flowed diagram $(D,\phi)$ of an $S^{1}$-oriented handlebody-link, we define the Alexander numbering of $(D,\phi)$ by assigning elements of $\mathbb{Z}_{m}$ to each region of $(D,\phi)$ as shown in Figure 7, where the unbounded region is labeled [math]. It is an extension of the Alexander numbering of a classical knot diagram [1]. It is easy to see that for any $\mathbb{Z}_{m}$-flowed diagram $(D,\phi)$ of an $S^{1}$-oriented handlebody-link, there uniquely exists the Alexander numbering of $(D,\phi)$. For example, a $\mathbb{Z}_{m}$-flowed diagram of the handlebody-knot $5_{2}$ [12] with the Alexander numbering is depicted in Figure 8. For any semi-arc $\alpha$ of $(D,\phi)$, we denote by $\rho(\alpha)$ the Alexander number of the region which the normal orientation of $\alpha$ points to.

In the following, every component of a diagram of any $S^{1}$-oriented handlebody-link may have a crossing at least $1$. Let $(D,\phi)$ be a $\mathbb{Z}_{m}$-flowed diagram of an $S^{1}$-oriented handlebody-link with the Alexander numbering and let $X$ be a $\mathbb{Z}_{m}$-family of Alexander biquandles. We put $C(D,\phi)=\{c_{1},\ldots,c_{n}\}$ and $V(D,\phi)=\{\tau_{1},\ldots,\tau_{2k}\}$, where $C(D,\phi)$ and $V(D,\phi)$ are the set of all crossings of $(D,\phi)$ and the one of all vertices of $(D,\phi)$ respectively, where the sign of $\tau_{i}$ is $1$ for any $i=1,\ldots,k$ and $-1$ for any $i=k+1,\ldots,2k$. Then we denote by $x_{i}$ each semi-arc of $(D,\phi)$ as shown in Figure 9, which implies $\mathcal{SA}(D,\phi)=\{x_{1},\ldots,x_{2n+3k}\}$.

We denote by $u_{i}$, $v_{i}$, $v^{\prime}_{i}$, $w_{i}$, $\alpha_{i}$, $\beta_{i}$ and $\gamma_{i}$ the semi-arcs incident to a crossing $c_{i}$ or a vertex $\tau_{i}$ as shown in Figure 10. We put $\phi_{i}:=\phi(u_{i})=\phi(w_{i})$, $\psi_{i}:=\phi(v_{i})=\phi(v^{\prime}_{i})$, $\eta_{i}:=\phi(\alpha_{i})$ and $\theta_{i}:=\phi(\beta_{i})$. We denote by $\epsilon_{c_{i}}\in\{\pm 1\}$ and $\epsilon_{\tau_{i}}\in\{\pm 1\}$ the signs of a crossing $c_{i}$ and a vertex $\tau_{i}$ respectively (see Figure 10).

For any semi-arcs $y,y^{\prime}\in\mathcal{SA}(D,\phi)$, we put

[TABLE]

Then we define a matrix $A(D,\phi;X)=(a_{i,j})\in M(2n+4k,2n+3k;X)$ by

[TABLE]

We note that $A(D,\phi;X)$ is ditermined up to permuting of rows and columns of the matrix, and it follows that

[TABLE]

For example, let $(E,\psi)$ be the $\mathbb{Z}_{m}$-flowed diagram of the handlebody-knot depicted in Figure 11. Then we have

[TABLE]

Then we have the following proposition.

Proposition 5.1**.**

Let $(D,\phi)$ be a $\mathbb{Z}_{m}$-flowed diagram of an $S^{1}$-oriented handlebody-link with the Alexander numbering and let $X$ be a $\mathbb{Z}_{m}$-family of Alexander biquandles. Let $\bm{a}_{i}$ be the $i$-th row of $A(D,\phi;X)$, that is,

[TABLE]

Then it follows that

[TABLE]

Proof.

For any semi-arc $y$ incident to a crossing or a vertex $\sigma$, we put

[TABLE]

It is sufficient to prove that for any $j=1,2,\ldots,2n+3k$,

[TABLE]

For the first term, we have

[TABLE]

For the second term, we have

[TABLE]

For the third term, we have

[TABLE]

For the last term, we have

[TABLE]

We note that

[TABLE]

Therefore for any $j=1,2,\ldots,2n+3k$, it follows that

[TABLE]

∎

Let $X$ be an Alexander biquandle and let $m=\operatorname{type}X$. Then $X$ is also a $\mathbb{Z}_{m}$-family of Alexander biquandles. Let $D$ be an oriented classical link diagram. We can regard $D$ as a $\mathbb{Z}_{m}$-flowed diagram $(D,\phi_{(1)})$ of an $S^{1}$-oriented handlebody-link whose components are of genus $1$, where $\phi_{(1)}$ is the constant map to $1$. Hence we can regard an $X$-coloring of $D$ as an $X$-coloring of $(D,\phi_{(1)})$. We define a matrix $A(D;X)\in M(2n,2n;X)$ by $A(D;X)=A(D,\phi_{(1)};X)$, where $n$ is the number of crossings of $D$. Then the set of all $X$-colorings of $D$, denoted by $\mathrm{Col}_{X}(D)$, is given by

[TABLE]

Therefore we obtain the following corollary.

Corollary 5.2**.**

Let $D$ be a diagram of an oriented classical link with the Alexander numbering and let $X$ be an Alexander biquandle. Let $\bm{a}_{i}$ be the $i$-th row of $A(D;X)$, that is,

[TABLE]

Then it follows that

[TABLE]

6. Main theorem

In this section, we give lower bounds for the Gordian distance and the unknotting number of $S^{1}$-oriented handlebody-knots.

Theorem 6.1**.**

Let $H_{i}$ be an $S^{1}$-oriented handlebody-knot of genus $g$ and let $D_{i}$ be a diagram of $H_{i}$ $(i=1,2)$. Let $X=\mathbb{Z}_{p}[t^{\pm 1}]/(f(t))$ which is a $\mathbb{Z}_{m}$-family of Alexander biquandles, where $p$ is a prime number, $s\in\mathbb{Z}_{p}[t^{\pm 1}]$ and $f(t)\in\mathbb{Z}_{p}[t^{\pm 1}]$ is an irreducible polynomial. Then it follows that

[TABLE]

Proof.

Let $(D,\phi)$ be a $\mathbb{Z}_{m}$-flowed diagram of an $S^{1}$-oriented handlebody-knot and let $C(D,\phi)=\{c_{1},\ldots,c_{n}\}$ and $V(D,\phi)=\{\tau_{1},\ldots,\tau_{2k}\}$. Let $(\overline{D},\overline{\phi})$ be the $\mathbb{Z}_{m}$-flowed diagram of an $S^{1}$-oriented handlebody-knot which is obtained from $(D,\phi)$ by the crossing change at $c_{1}$ and let $C(\overline{D},\overline{\phi})=\{\overline{c}_{1},\ldots,\overline{c}_{n}\}$ and $V(\overline{D},\overline{\phi})=\{\overline{\tau}_{1},\ldots,\overline{\tau}_{2k}\}$, where $\overline{\phi}$, $\overline{c}_{i}$ and $\overline{\tau}_{i}$ originate from $\phi$, $c_{i}$ and $\tau_{i}$ naturally and respectively (see Figure 12). In the following, we show that

[TABLE]

that is,

[TABLE]

We may assume that $c_{1}$ is a positive crossing and $\overline{c}_{1}$ is a negative crossing. We denote by $\overline{x}_{i}$ each semi-arc of $(\overline{D},\overline{\phi})$ in the same way as in Figure 9 with respect to $\overline{c}_{i}$ or $\overline{\tau}_{i}$, and so are $\overline{v}^{\prime}_{i}$, $\overline{w}_{i}$, $\overline{\alpha}_{i}$, $\overline{\beta}_{i}$, $\overline{\phi}_{i}$, $\overline{\psi}_{i}$, $\overline{\eta}_{i}$, $\overline{\theta}_{i}$, $\overline{\epsilon}_{c_{i}}$ and $\overline{\epsilon}_{\tau_{i}}$ (see Figure 10). We denote by $x_{j_{1}}$ and $x_{j_{2}}$ the semi-arcs which point to the crossing $c_{1}$ of $(D,\phi)$ as shown in Figure 12, and we put $a:=\phi_{1}=\overline{\psi}_{1}$ and $b:=\psi_{1}=\overline{\phi}_{1}$. We note that $\mathrm{Col}_{X}(D,\phi)$ and $\mathrm{Col}_{X}(\overline{D},\overline{\phi})$ are vector spaces over $X$ since $X$ is a field.

Let

[TABLE]

and let

[TABLE]

where $\bm{\hat{a}}_{i}$ is a vector obtained by permuting the first entry and the $(n+1)$-th entry of $\bm{\overline{a}}_{i}$. Then we have $\bm{a}_{i}=\bm{\hat{a}}_{i}$ when $i\neq 1,n+1$. We note that $\operatorname{rank}A(\overline{D},\overline{\phi};X)=\operatorname{rank}\hat{A}(\overline{D},\overline{\phi};X)$ and

[TABLE]

By Proposition 5.1, we obtain

[TABLE]

and

[TABLE]

If $\epsilon_{c_{1}}t^{-\rho(w_{1})}(s^{\phi_{1}}-t^{\phi_{1}})=0$, we have $s^{\phi_{1}}-t^{\phi_{1}}=s^{a}-t^{a}=0$, which implies that $\bm{a}_{n+1}=-\bm{\hat{a}}_{1}$. Hence it follows that

[TABLE]

If $\overline{\epsilon}_{c_{1}}t^{-\rho(\overline{w}_{1})}(s^{\overline{\phi}_{1}}-t^{\overline{\phi}_{1}})=0$, we have $s^{\overline{\phi}_{1}}-t^{\overline{\phi}_{1}}=s^{b}-t^{b}=0$, which implies that $\bm{a}_{1}=-\bm{\hat{a}}_{n+1}$. Hence it follows that

[TABLE]

If $\epsilon_{c_{1}}t^{-\rho(w_{1})}(s^{\phi_{1}}-t^{\phi_{1}})\neq 0$ and $\overline{\epsilon}_{c_{1}}t^{-\rho(\overline{w}_{1})}(s^{\overline{\phi}_{1}}-t^{\overline{\phi}_{1}})\neq 0$, we can represent $\bm{a}_{1}$ and $\bm{\overline{a}}_{1}$ as linear combinations of $\bm{a}_{2},\ldots,\bm{a}_{2n+4k}$ and $\bm{\overline{a}}_{2},\ldots,\bm{\overline{a}}_{2n+4k}$ respectively. Hence it follows that

[TABLE]

which implies that

[TABLE]

Consequently, if we can deform $H_{1}$ into $H_{2}$ by crossing changes at $l$ crossings, then for any $\mathbb{Z}_{m}$-flowed diagram $(D_{1},\phi_{1})$ of $H_{1}$, there exists a $\mathbb{Z}_{m}$-flowed diagram $(D_{2},\phi_{2})$ of $H_{2}$ satisfying $\gcd\phi_{1}=\gcd\phi_{2}$ and

[TABLE]

by Lemma 4.1. Therefore it follows that

[TABLE]

∎

By Proposition 4.3 and Theorem 6.1, the following corollary holds immediately.

Corollary 6.2**.**

Let $H$ be an $S^{1}$-oriented handlebody-knot and let $D$ be a diagram of $H$. Let $X=\mathbb{Z}_{p}[t^{\pm 1}]/(f(t))$ which is a $\mathbb{Z}_{m}$-family of Alexander biquandles, where $p$ is a prime number, $s\in\mathbb{Z}_{p}[t^{\pm 1}]$ and $f(t)\in\mathbb{Z}_{p}[t^{\pm 1}]$ is an irreducible polynomial. Then it follows that

[TABLE]

7. Examples

In this section, we give some examples. In Example 7.1, we give a handlebody-knot with unknotting number $2$, and in Remark 7.2, we note that it can not be obtained by using Alexander quandle colorings with $\mathbb{Z}_{2},\mathbb{Z}_{3}$-flows introduced in [14]. In Example 7.3, we give three handlebody-knots with unknotting number $n$ for any $n\in\mathbb{Z}_{>0}$. In Example 7.4, we give two handlebody-knots with their Gordian distance $n$ for any $n\in\mathbb{Z}_{>0}$.

Example 7.1**.**

Let $H$ be the handlebody-knot represented by the $\mathbb{Z}_{10}$-flowed diagram $(D,\phi)$ depicted in Figure 13. Then we show that $u(H)=2$.

Let $s=1\in\mathbb{Z}_{3}[t^{\pm 1}]$ and let $f(t)=t^{4}+2t^{3}+t^{2}+2t+1\in\mathbb{Z}_{3}[t^{\pm 1}]$, which is an irreducible polynomial. Then $X:=\mathbb{Z}_{3}[t^{\pm 1}]/(f(t))$ is a $\mathbb{Z}_{10}$-family of Alexander biquandles. Then for any $x,y,z\in X$, the assignment of them to each semi-arc of $(D,\phi)$ as shown in Figure 13 is an $X$-coloring of $(D,\phi)$, which implies $\dim\mathrm{Col}_{X}(D,\phi)\geq 3$. By Corollary 6.2, we obtain $2\leq u(H)$. On the other hand, we can deform $H$ into a trivial handlebody-knot by the crossing changes at two crossings surrounded by dotted circles depicted in Figure 13. Therefore it follows that $u(H)=2$.

Remark 7.2**.**

We show that the result in Example 7.1 can not be obtained by using Alexander quandle colorings with $\mathbb{Z}_{2},\mathbb{Z}_{3}$-flows introduced in [14].

Let $H$ be the handlebody-knot represented by the $\mathbb{Z}_{m}$-flowed diagram $(D,\phi(a,b))$ depicted in Figure 14 for any $m=2,3$ and $a,b\in\mathbb{Z}_{m}$. Let $p$ be a prime number, $s=1\in\mathbb{Z}_{p}[t^{\pm 1}]$, $f(t)$ be an irreducible polynomial in $\mathbb{Z}_{p}[t^{\pm 1}]$ and let $X=\mathbb{Z}_{p}[t^{\pm 1}]/(f(t))$ which is a $\mathbb{Z}_{m}$-family of Alexander (bi)quandles. We note that $\mathrm{Col}_{X}(D,\phi(a,b))$ is generated by $x,y,z\in X$ as shown in Figure 14 for any $m=2,3$ and $a,b\in\mathbb{Z}_{m}$. If $(a,b)=(1,0)$, $x$, $y$ and $z$ need to satisfy the following relations:

[TABLE]

that is,

[TABLE]

where

[TABLE]

These relations are obtained from crossings $c_{1},c_{2},c_{3}$ and $c_{4}$ as shown in Figure 14. When $t^{2}-t+1\neq 0$ in $X$, it is clearly that $\operatorname{rank}M\geq 1$. When $t^{2}-t+1=0$ in $X$, we have

[TABLE]

which implies that $\operatorname{rank}M=1$. Hence we have $\dim\mathrm{Col}_{X}(D,\phi(1,0))=3-\operatorname{rank}M\leq 2$. Therefore we can not obtain $2\leq u(H)$.

We can prove the remaining cases in the same way.

Example 7.3**.**

Let $A_{n}$, $B_{n}$ and $C_{n}$ be the handlebody-knots represented by the $\mathbb{Z}_{8}$-flowed diagram $(D_{A_{n}},\phi_{A_{n}})$, the $\mathbb{Z}_{24}$-flowed diagram $(D_{B_{n}},\phi_{B_{n}})$ and the $\mathbb{Z}_{8}$-flowed diagram $(D_{C_{n}},\phi_{C_{n}})$ depicted in Figure 15, 16 and 17 respectively for any $n\in\mathbb{Z}_{>0}$. Then we show that $u(A_{n})=u(B_{n})=u(C_{n})=n$.

1. (1)

Let $s=t+1\in\mathbb{Z}_{3}[t^{\pm 1}]$ and let $f(t)=t^{2}+t+2\in\mathbb{Z}_{3}[t^{\pm 1}]$, which is an irreducible polynomial. Then $X:=\mathbb{Z}_{3}[t^{\pm 1}]/(f(t))$ is a $\mathbb{Z}_{8}$-family of Alexander biquandles. Then for any $x_{0},x_{1},\ldots,x_{n}\in X$, the assignment of them to each semi-arc of $(D_{A_{n}},\phi_{A_{n}})$ as shown in Figure 15 is an $X$-coloring of $(D_{A_{n}},\phi_{A_{n}})$, which implies $\dim\mathrm{Col}_{X}(D_{A_{n}},\phi_{A_{n}})\geq n+1$. By Corollary 6.2, we obtain $n\leq u(A_{n})$. On the other hand, we can deform $A_{n}$ into a trivial handlebody-knot by the crossing changes at $n$ crossings surrounded by dotted circles depicted in Figure 15. Therefore it follows that $u(A_{n})=n$.

2. (2)

Let $s=t^{2}+1\in\mathbb{Z}_{5}[t^{\pm 1}]$ and let $f(t)=t^{2}+2t+4\in\mathbb{Z}_{5}[t^{\pm 1}]$, which is an irreducible polynomial. Then $X:=\mathbb{Z}_{5}[t^{\pm 1}]/(f(t))$ is a $\mathbb{Z}_{24}$-family of Alexander biquandles. Then for any $x_{0},x_{1},\ldots,x_{n}\in X$, the assignment of them to each semi-arc of $(D_{B_{n}},\phi_{B_{n}})$ as shown in Figure 16 is an $X$-coloring of $(D_{B_{n}},\phi_{B_{n}})$, which implies $\dim\mathrm{Col}_{X}(D_{B_{n}},\phi_{B_{n}})\geq n+1$. By Corollary 6.2, we obtain $n\leq u(B_{n})$. On the other hand, we can deform $B_{n}$ into a trivial handlebody-knot by the crossing changes at $n$ crossings surrounded by dotted circles depicted in Figure 16. Therefore it follows that $u(B_{n})=n$.

3. (3)

Let $s=2t-1\in\mathbb{Z}_{3}[t^{\pm 1}]$ and let $f(t)=t^{2}+t+2\in\mathbb{Z}_{3}[t^{\pm 1}]$, which is an irreducible polynomial. Then $X:=\mathbb{Z}_{3}[t^{\pm 1}]/(f(t))$ is a $\mathbb{Z}_{8}$-family of Alexander biquandles. Then for any $x_{0},x_{1},\ldots,x_{n}\in X$, the assignment of them to each semi-arc of $(D_{C_{n}},\phi_{C_{n}})$ as shown in Figure 17 is an $X$-coloring of $(D_{C_{n}},\phi_{C_{n}})$, which implies $\dim\mathrm{Col}_{X}(D_{C_{n}},\phi_{C_{n}})\geq n+1$. By Corollary 6.2, we obtain $n\leq u(C_{n})$. On the other hand, we can deform $C_{n}$ into a trivial handlebody-knot by the crossing changes at $n$ crossings surrounded by dotted circles depicted in Figure 17. Therefore it follows that $u(C_{n})=n$.

Example 7.4**.**

Let $H_{n}$ and $H^{\prime}_{n}$ be the handlebody-knots represented by the $\mathbb{Z}_{3}$-flowed diagrams $(D_{n},\phi_{n})$ and $(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))$ respectively depicted in Figure 18 for any $n\in\mathbb{Z}_{>0}$ and $a,b\in\mathbb{Z}_{3}$. Then we show that $d(H_{n},H^{\prime}_{n})=n$.

Let $s=1\in\mathbb{Z}_{2}[t^{\pm 1}]$ and let $f(t)=t^{2}+t+1\in\mathbb{Z}_{2}[t^{\pm 1}]$, which is an irreducible polynomial. Then $X:=\mathbb{Z}_{2}[t^{\pm 1}]/(f(t))$ is a $\mathbb{Z}_{3}$-family of Alexander (bi)quandles. Then for any $x_{0},x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\in X$, the assignment of them to each semi-arc of $(D_{n},\phi_{n})$ as shown in Figure 18 is an $X$-coloring of $(D_{n},\phi_{n})$, which implies $\dim\mathrm{Col}_{X}(D_{n},\phi_{n})\geq 2n+1$.

On the other hand, we note that $\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))$ is generated by $x_{0},x_{1},x^{\prime}_{1},\ldots,x_{n},$ $x^{\prime}_{n},y_{1},y^{\prime}_{1},\ldots,y_{n},y^{\prime}_{n}\in X$ as shown in Figure 18 for any $a,b\in\mathbb{Z}_{3}$. If $(a,b)=(0,0)$, it is easy to see that $\dim\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))=1$. If $(a,b)=(1,1),(1,2),(2,1),(2,2)$, we obtain that $x_{i}=x^{\prime}_{i}=y_{i}=y^{\prime}_{i}$ for any $i=1,2,\ldots,n$, which implies $\dim\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))\leq n+1$. If $(a,b)=(0,1),(0,2)$, we have

[TABLE]

Hence $\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))$ is generated by $x_{0},y_{1},\ldots,y_{n}\in X$, which implies $\dim\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))\leq n+1$. If $(a,b)=(1,0),(2,0)$, in the same way as when $(a,b)=(0,1),(0,2)$, $\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))$ is generated by $x_{0},x_{1},\ldots x_{n}\in X$, which implies $\dim\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))\leq n+1$. Hence for any $a,b\in\mathbb{Z}_{3}$, $\dim\mathrm{Col}_{X}(D^{\prime}_{n},\phi^{\prime}_{n}(a,b))\leq n+1$, which implies that

[TABLE]

By Theorem 6.1, it follows that $n\leq d(H_{n},H^{\prime}_{n})$.

Finally, we can deform $H^{\prime}_{n}$ into $H_{n}$ by the crossing changes at $n$ crossings surrounded by dotted circles depicted in Figure 18. Therefore it follows that $d(H_{n},H^{\prime}_{n})=n$.

Acknowledgment

The author would like to thank Masahide Iwakiri for his helpful comments. He is particularly grateful to Atsushi Ishii for invaluable advice and his suggestions.