Virtualized Delta moves for virtual knots and links

This follows from Figure 2.1, where the symbol $\overset{\textrm{R}}{\longleftrightarrow}$ means a combination of generalized Reidemeister moves. ∎

This follows from Figure 2.3. ∎

The sequence of Figure 2.4 shows that an upper forbidden move is realized by a combination of two crossing changes and a forbidden detour move, where the symbol $\overset{\textrm{cc}}{\longleftrightarrow}$ means a combination of crossing changes. By Lemmas 2.1 and 2.2, we have the result for the case of an upper forbidden move. The case of a lower forbidden move is proved similarly. ∎

Since the forbidden move is an unknotting operation for virtual knots [6, 8], we have the conclusion by Lemma 2.4. ∎

Let $G$ and $G^{\prime}$ be Gauss diagrams of $K$ and $K^{\prime}$, respectively. It is sufficient to prove that if $G$ and $G^{\prime}$ are related by a virtualized $\Delta$-move, then the odd writhes satisfy $J(K)-J(K^{\prime})\in\{0,\pm 2\}$.

We may assume that $G^{\prime}$ is obtained from $G$ by removing three chords corresponding to three real crossings involved in a virtualized $\Delta$-move. See Figure 2.5. It is known that among these three chords, the number of chords with index odd is equal to zero or two (cf. [7, 10]). Furthermore, the parity of the index of any other chord is preserved by the virtualized $\Delta$-move; in fact, each pair of the three chords has two adjacent endpoints on the underlying circle. See the figure again. Therefore we have the conclusion. ∎

For an integer $s\geq 1$, we consider a knot diagram and its Gauss diagram with $2m+2s-1$ real crossings (or chords) as shown in Figure 2.6. Let $K_{s}(m)$ be the virtual knot presented by this diagram.

Since $K_{s}(m)$ can be unknotted by $m$ crossing changes at $m$ real crossings among $2m$ half twists in the knot diagram, we have ${\rm u}(K_{s}(m))\leq m$. By Lemma 2.7, we have

On the other hand, the $2m$ horizontal chords of the Gauss diagram have the sign $-1$ and indices $\pm 1$, and the $2s-1$ vertical chords have index $0$. Since $J(K_{s}(m))=-2m$ holds, we have ${\rm u}_{v\Delta}(K_{s}(m))\geq m$ by Proposition 2.6(ii), and hence ${\rm u}_{v\Delta}(K_{s}(m))={\rm u}(K_{s}(m))=m$.

Now it is enough to prove that the virtual knots $K_{s}(m)$’s are mutually distinct. By a straightforward calculation, the Jones polynomial $f_{K_{s}(m)}(A)\in{\mathbb{Z}}[A,A^{-1}]$ of $K_{s}(m)$ is given by

Since the maximal degree of $f_{K_{s}(m)}(A)$ is equal to $8m+8s-6$, we have $K_{s}(m)\neq K_{s^{\prime}}(m)$ for any $s\neq s^{\prime}$. ∎

For an integer $s\geq 2$, we consider a long virtual knot diagram and its Gauss diagram with $2s+3$ real crossings (or chords) $a_{1},a_{2},a_{3}$ and $b_{1},\dots,b_{2s}$ as shown in Figure 2.7. Let $T_{s}$ be the long virtual knot presented by this diagram, and $K_{s}(m)$ the virtual knot obtained from the closure of the product of $m$ copies of $T_{s}$.

Since $T_{s}$ can be unknotted by a single virtualized $\Delta$-move involving the three crossings $a_{1}$, $a_{2}$, and $a_{3}$ in the long knot diagram, we have ${\rm u}_{v\Delta}(K_{s}(m))\leq m$.

On the other hand, since we have

it follows from [9, Lemma 4.3] that

This induces $J(K_{s}(m))=J_{1}(K_{s}(m))+J_{-2s-1}(K_{s}(m))=2m$. Therefore we have ${\rm u}_{v\Delta}(K_{s}(m))\geq m$ by Proposition 2.6(ii), and hence ${\rm u}_{v\Delta}(K_{s}(m))=m$. Furthermore, since $J_{1}(K_{s}(m))=m\neq 0=J_{-1}(K_{s}(m))$ holds, we have ${\rm u}(K_{s}(m))=\infty$ by [9, Theorem 1.5].

Since $J_{2}(K_{s}(m))=2ms$ holds, we have $K_{s}(m)\neq K_{s^{\prime}}(m)$ for any $s\neq s^{\prime}$. ∎

This follows from Figure 3.2. ∎

(i) This follows from Lemma 3.1.

(ii) Let $G^{\prime\prime\prime}$ be a Gauss diagram obtained from $G$ by reversing the orientation and changing the sign of $\gamma$. By Lemma 2.1, $G$ and $G^{\prime\prime\prime}$ are related by a finite sequence of a virtualized $\Delta$-move and Reidemeister moves. Furthermore $G^{\prime\prime}$ is obtained from $G^{\prime\prime\prime}$ by reversing the orientation of $\gamma$. Therefore we have the conclusion by (i). ∎

This follows from Lemmas 2.2 and 2.4. ∎

Let $G=\bigcup_{i=1}^{n}G_{i}$ be a Gauss diagram of $L$. By using Lemma 3.3 and Reidemeister moves I, we can remove all the self-chords from $G$. Therefore we may assume that $G$ has no self-chords (up to virtualized $\Delta$-moves and Reidemeister moves).

If there is a nonself-chord between $G_{i}$ and $G_{j}$ $(2\leq i<j\leq n)$, then we can replace it with a pair of nonself-chords one of which connects between $G_{1}$ and $G_{i}$, and the other between $G_{1}$ and $G_{j}$. In fact, this is achieved by observing a sequence of link diagrams as shown in Figure 3.4. Therefore we may also assume that there are no nonself-chords between $G_{i}$ and $G_{j}$ $(2\leq i<j\leq n)$.

By using Lemmas 3.2, 3.3, and Reidemeister moves II, we can reduce the number of nonself-chords between $G_{1}$ and $G_{i}$ $(2\leq i\leq n)$ to zero or one.

Finally, any Gauss diagram $G$ of $L$ can be deformed into $H(a_{2},\dots,a_{n})$ for some $a_{i}$’s by Lemmas 3.2 and 3.3. ∎

Since the chord(s) involved in a Reidemeister move or a virtualized $\Delta$-move have an even number of endpoints on each $G_{i}$, this move preserves $p_{i}(L)$. ∎

The only if part follows from Lemma 3.5.

We will prove the if part. By Proposition 3.4, $L$ and $L^{\prime}$ are $v\Delta$-equivalent to $M(a_{2},\dots,a_{n})$ and $M(b_{2},\dots,b_{n})$ for some $a_{i}$’s and $b_{i}$’s, respectively. By assumption and Lemma 3.5, it holds that

for any $i=1,\dots,n$. Therefore $L$ and $L^{\prime}$ are $v\Delta$-equivalent to $M(a_{2},\dots,a_{n})=M(b_{2},\dots,b_{n})$. ∎

For $X=rv\Delta$, a crossing change at a real crossing is obtained from the deformation sequence in Figure 2.1 by replacing $\stackrel{{\scriptstyle v\Delta}}{{\longleftrightarrow}}$ with $\stackrel{{\scriptstyle rv\Delta}}{{\longrightarrow}}$. For $X=vr\Delta$, we may follow this sequence in reverse. For $X=rv\Delta^{\prime}$ and $vr\Delta^{\prime}$, we may perform the crossing change at every real crossing in the above sequences for $rv\Delta$ and $vr\Delta$, respectively. ∎

Figure 4.3 indicates the proof. More precisely, the first deformation in this figure is obtained from the deformation sequence in Figures 3.2 by replacing $\stackrel{{\scriptstyle v\Delta}}{{\longleftrightarrow}}$ with $\stackrel{{\scriptstyle rv\Delta}}{{\longrightarrow}}$. The second deformation is a crossing change which is realized by an $rv\Delta$-move and generalized Reidemeister moves by Lemma 4.1. ∎

We use the notation “$X\Rightarrow Y$” if a $Y$-move is realized by a combination of $X$-moves and generalized Reidemeister moves. Then we have the following.

• •

  $rv\Delta\Rightarrow vr\Delta$: The sequence in Figure 4.4 shows that a $vr\Delta$-move is realized by a combination of an $rv\Delta$-move, three sign reversal moves, and several generalized Reidemeister moves. Therefore we have $rv\Delta\Rightarrow vr\Delta$ by Lemma 4.2.

• •

  $vr\Delta\Rightarrow vr\Delta^{\prime}$: The sequence in Figure 4.5 shows that a $vr\Delta^{\prime}$-move is realized by a $vr\Delta$-move and three crossing changes. Therefore we have $vr\Delta\Rightarrow vr\Delta^{\prime}$ by Lemma 4.1.

• •

  $vr\Delta^{\prime}\Rightarrow rv\Delta^{\prime}$: We may follow any sequence for $rv\Delta\Rightarrow vr\Delta$ with opposite crossing information in reverse.

• •

  $rv\Delta^{\prime}\Rightarrow rv\Delta$: We may follow any sequence for $vr\Delta\Rightarrow vr\Delta^{\prime}$ with opposite crossing information in reverse.

Therefore we have the cycle

which induces the other eight cases immediately. ∎