ON INEQUALITIES BETWEEN UNKNOTTING NUMBERS AND CROSSING NUMBERS OF SPATIAL EMBEDDINGS OF TRIVIALIZABLE GRAPHS AND HANDLEBODY-KNOTS

Let $L$ be a link in the 3-dimensional Euclidean space $\mathbb{R}^{3}$. The unknotting number $u(L)$ is the minimal number of crossing changes (Fig. 1) from $L$ to a trivial link. The crossing number $c(L)$ is the minimal number of crossing points among all regular diagrams of $L$. It is well-known that $u(L)$ is less than or equal to half of $c(L)$ (see for example [12]). In [12] Taniyama characterized the links which satisfy the equality as follows.

In [1] Taniyama and the author showed that this inequality is not extended to spatial embeddings of planar graphs and this inequality is extended to spatial embeddings of trivializable planar graphs. Namely for any spatial embedding $f$ of a trivializable planar graph, $u(f)$ is less than or equal to half of $c(f)$. For example, a handcuff-graph and a theta-curve as illustrated in Fig. 2 are trivializable. We characterize the spatial embeddings of a handcuff-graph or a theta curve which satisfy the equality as follows.

We note that unknotting numbers of spatial embeddings of a theta curve is studied in [2].

A handlebody-knot is an embedded handlebody in the 3-dimensional Euclidean space $\mathbb{R}^{3}$, which is introduced by Ishii in [5]. Two handlebody-knots $H_{1}$ and $H_{2}$ are equivalent if there is an orientation-preserving homeomorphism $h$ of $\mathbb{R}^{3}$ with $h(H_{1})=H_{2}$. A spine of a handlebody-knot $H$ is a spatial graph whose regular neightborhood is $H$. In this paper, we assume that spines have no degree 1 verticies. Any handlebody-knot $H$ can be represented by a spatial trivalent graph that is a spine of $H$. In particular, genus 2 handlebody-knot can be represented by a spatial embedding of a handcuff-graph or a theta curve. A crossing change of a handlebody-knot $H$ is that of a spatial trivalent graph representing $H$. In [6] Iwaliri showed that a crossing change of a handlebody-knot is an unknotting operation and give lower bounds of the unknotting numbers for handlebody-knots by the numbers of some finite Alexander quandle colorings.

We have the following well-known relation between unknotting number and crossing number of classical knots.

In [12] Taniyama characterized the knots which satisfy the equality as follows.

In this paper, as an extension of Proposition 1.4, we show the following theorem.

The spine of genus 1 handlebody-knot is a classical knot. Therefore Theorem 1.6 is an extention of Proposition 1.4. It follows from Theorem 1.2 and Theorem 1.3 that for any non-trivial handlebody-knot $H$ with genus 2 twice the unknotting number of $H$ is less than or equal to the crossing number of $H$ minus one  (see section 4).

It follows from Theorem 1.5 that genus 1 handlebody-knot $H$ with $u(H)=\dfrac{c(H)-1}{2}$ is a regular neighborhood of a $(2,\ p)-$torus knot. We also characterize genus $n\geq 2$ handlebody-knots which satisfy the equality as follows.

This paper consists of five sections. In section 2 we review trivializability of planar graphs and inequalities between unknotting numbers and crossing numbers of spatial embeddings of planar graphs. In section 3 we introduce unknotting number of handlebody-knots. In section 4 we give proofs of Theorem 1.2 and Theorem 1.3. In section 5 we give proofs of Theorem 1.6 and Theorem 1.7.

Let $G$ be a planar graph. A spatial embedding of $G$ is an embedding $f:G\rightarrow\mathbb{R}^{3}$. Its image $f(G)$ is said to be a spatial graph. Let $\pi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$ be a natural projection defined by $\pi(x,y,z)=(x,y)$. Let $SE(G)$ be the set of all spatial embeddings of $G$. A regular projection of $G$ is a continuous map $\tilde{f}:G\rightarrow\mathbb{R}^{2}$ whose double points are only finitely many transversal double points. Such a double point is said to be a crossing point or simply a crossing. If we give over/under informations at each crossing points of a regular projection $\tilde{f}$ of $G$, then $\tilde{f}$ together with the over/under informations represents a spatial embedding $f:G\rightarrow\mathbb{R}^{3}$ such that $\tilde{f}=\pi\circ f$. Such a regular projection together with the over/under informations is said to be a diagram of $f(G)$. Then we say that $f$ is obtained from $\tilde{f}$. We also call $\tilde{f}$ a regular projection of $f(G)$. For a diagram $D$ of a spatial embedding, the set of all crossings of $D$ is denoted by $\textit{C}(D)$. The number of crossings of $D$ is denoted by $c(D)=|\textit{C}(D)|$.

An element $f\in SE(G)$ is said to be trivial, if it is ambient isotopic to $t\in SE(G)$ such that $t(G)\subset\mathbb{R}^{2}$. Any spatial embedding of a planar graph can be transformed into trivial one by crossing changes. Therefore unknotting number is naturally extended to spatial embeddings of planar graphs as follows. For $f\in SE(G)$, the unknotting number $u(f)$ is defined to be the minimal number of crossing changes from $f$ to a trivial embedding of $G$. The crossing number $c(f)$ is defined to be the minimal number of crossing points among all diagrams of spatial embeddings that are ambient isotopic to $f$.

For any link $L$, $L$ satisfies the inequality $u(L)\leq\dfrac{c(L)}{2}$. But this is not extended for spatial embeddings of planar graph, namely there are a planar graph $G$ and a spatial embedding $f$ of $G$ such that $u(f)>\dfrac{c(f)}{2}$. Let $P_{3}$ the cube graph and $f_{3}\in SE(P_{3})$ a spatial embedding of $P_{3}$ as illustrated in Fig. 4. The spatial graph $f_{3}(P_{3})$ contains three Hopf-links and one crossing change of edges of $f_{3}(P_{3})$ unknot at most two of them (See Fig. 5). Then $u(f_{3})\geq 2$. Since $f_{3}(P_{3})$ contains a trefoil whose crossing number is 3, $c(f_{3})=3$ and $u(f_{3})>\dfrac{c(f_{3})}{2}$ [1].

Now we review the reason why it happens for some planar graphs. The key point of the proof of $u(L)\leq\dfrac{c(L)}{2}$ for a link $L$ is that any link diagram can be transformed into a trivial link diagram by changing over/under informations at some crossings of the diagram. Let $D$ be a minimal crossing diagram of $L$. Let $A$ be a subset of $C(D)$ such that changing over/under informations at all crossings in $A$ turns $D$ to a diagram $T_{1}$ of a trivial link. Let $T_{2}$ be a diagram that is obtained from $T_{1}$ by changing over/under informations at all crossings. A mirror image of a trivial link is also trivial. Thus $T_{2}$ is a diagram of a trivial link. Note that $T_{2}$ is obtained from $D$ by changing over/under informations at all crossings in $C(D)-A$. Therefore we have

On the other hand, all diagrams obtained from $\pi\circ f_{3}(P_{3})$ (Fig. 6) represent non-trivial spatial graphs since each of the spatial graphs obtained from these diagrams contains at least one Hopf-link. A regular projection $\tilde{f}$ of a planar graph $G$ is said to be a knotted projection [11], if all spatial embeddings of $G$ which can be obtained from $\tilde{f}$ are non-trivial.

A planar graph is said to be trivializable if it has no knotted projections. In [11] Taniyama gave a class of trivializable graphs. In [9] Sugiura and Suzuki extended the class. In [10] Tamura gave another class of trivializable graphs.

For a spatial embedding of a trivializable planar graph, the same argument as for a link works, and we have the following proposition.

We review that crossing change of a handlebody-knot is an unknotting operation [6].

A diagram of a handlebody-knot $H$ is that of a spatial trivalent graph representing $H$. In [5], Ishii gave a list of fundamental moves among diagrams of handlebody-knots, which is called R1-6 moves illustrated in Fig. 7. Ishii showed that two handlebody-knots are equivalent if and only if their representing diagrams are related by a finite sequence of R1-6 moves. Note that R6-move is also called IH-move.

A crossing change of a handlebody-knot $H$ is that of a spatial trivalent graph representing $H$. This move can be realized by switching two tubes illustrated in Fig. 8. A genus $n$ handlebody-knot is trivial if it is equivalent to a handlebody-knot represented by a diagram illustrated in Fig. 9.

Let $T_{n}$ be the trivalent graph whose image is illustrated in Fig. 9. Any handlebody-knot is represented by a diagram of a spatial embedding of $T_{n}$ since a genus $n$ handlebody has $T_{n}$ as a spine. Note that $T_{n}$ is a trivializable graph [9]. Namely, any diagram $D$ of a spatial embedding of $T_{n}$ can be changed to a trivial spatial graph diagram by changing over/under informations at some crossings of $D$. Then we have the following proposition.

Therefore unknotting number is naturally extended to handlebody-knots as follows. For a handlebody-knot $H$, the unknotting number $u(H)$ is the minimal number of crossing changes needed to obtain a trivial handlebody-knot from $H$. The crossing number $c(H)$ is the minimal number of crossing points among all diagrams of handlebody-knots that are equivalent to $H$.

By the proof of [8, Proposition 3.1] we see that any diagram $D$ of a spatial graph can be transformed into a diagram of a spatial graph whose neighborhood are ambient isotopic to a neighborhood of a trivial bouquet by changing over/under informations at some crossings of $D$. Therefore, in [6], Iwakiri also showed that Proposition 3.1 can be refined to the strong statement as follows.

For a handlebody-knot diagram $D$, the unknotting number $u(D)$ is the minimal number of changing over/under informations at crossings of $D$ needed to obtain a trivial handlebody-knot diagram. Same as Proposition 2.1 we have $u(D)\leq\dfrac{c(D)}{2}$ and $u(H)\leq\dfrac{c(H)}{2}$.

In section 5, we show that a handlebody-knot $H$ satisfies $u(H)=\dfrac{c(H)}{2}$ if and only if $H$ is trivial (Theorem 1.6). Then it is natural to ask when handlebody-knots satisfy the equality $u(H)=\dfrac{c(H)-1}{2}$. Let $H$, $H_{1}$ and $H_{2}$ be handlebody-knots in $\mathbb{R}^{3}$ and let $S$ be a $2-$sphere in $\mathbb{R}^{3}$. Suppose that $H\cap S=H_{1}\cap H_{2}$ is a $2-$disk and $H=H_{1}\cup H_{2}$. Then $H$ is said to be a disk sum of $H_{1}$ and $H_{2}$ and denoted by $H=H_{1}\#H_{2}$. In [12] Taniyama showed that if a classical knot $K$ satisfies $u(K)=\dfrac{c(K)-1}{2}$ then $K$ is a $(2,\ p)$-torus knot for some odd number $p\neq\pm 1$ (Theorem 1.5). Therefore the handlebody-knots illustrated in Fig. 10 may satisfy the equality. But by the following proposition only two of these handlebody-knots satisfy the equality.

In this section we prove Theorem 1.6 and Theorem 1.7. In the following we give an inequality between unknotting number and crossing number by an observation of subdivided graph.

Let $\tilde{f}:G\rightarrow\mathbb{R}^{2}$ be a regular projection of a graph $G$. Let $c_{1},\ c_{2},\ \cdots,\ c_{k}$ be crossing points of $\tilde{f}(G)$. A subdivided graph of $G$ at $\{c_{1},\ c_{2},\ \cdots,\ c_{k}\}$ is a graph obtained by adding $2k$ vertices $v_{1},\ v_{1}^{\prime},\ v_{2},\ v_{2}^{\prime},\ \cdots,v_{k},\ v_{k}^{\prime}$ to $G$ such that $\tilde{f}(v_{i})=\tilde{f}(v_{i}^{\prime})=c_{i}$ and $v_{i}$ (resp. $v_{i}^{\prime}$) is contained in the over-arc (resp. the under-arc) at $c_{i}$. Then we say that $v_{i}$ (resp. $v_{i}^{\prime})$ is an over-vertex (resp. under-vertex) at $c_{i}$. Let $T$ be a spanning tree of $G^{\prime}$. For any two vertices $v$ and $u$ of $G^{\prime}$, let $d_{T}(v,u)$ be the number of edges of the path in $T$ joining $v$ and $u$.

By subdividing $T$ if necessary, we can choose a vertex $v$ of $T$ such that $d_{T}(v,v_{1})=d_{T}(v,v_{1}^{\prime})$ since there exists a path in $T$ joining $v_{1}$ and $v_{1}^{\prime}$ (see for example Fig. 25). Then by Lemma 5.1 we have $u(D)\leq\dfrac{c(D)-1}{2}$. Since $u(H)\leq u(D)$ and $c(D)=c(H)$ we have $u(H)\leq\dfrac{c(H)-1}{2}$. ∎