Virtualized Delta, sharp, and pass moves for oriented virtual knots and links

(i) The sequence in the top row of Figure 2.2 shows that a crossing change is realized by a combination of a $v\Delta_{1}^{\wedge}$-move and several generalized Reidemeister moves, where the symbol $\stackrel{{\scriptstyle\rm R}}{{\longleftrightarrow}}$ means a combination of generalized Reidemeister moves. For a $v\Delta_{2}^{\wedge}$-move, we may use the above sequence with the orientations of all the strings reversed. See the second row of the figure. For $v\Delta_{3}^{\wedge}$- and $v\Delta_{4}^{\wedge}$-moves, we may use the sequences for $v\Delta_{1}^{\wedge}$- and $v\Delta_{2}^{\wedge}$-moves with opposite crossing information at every real crossing, respectively. See the third and bottom rows of the figure.

(ii) The sequence in Figure 2.3 shows that a crossing change is realized by a combination of a $v\Delta_{1}^{\circ}$-move and several generalized Reidemeister moves. We remark that it is obtained from the sequence for a $v\Delta_{1}^{\wedge}$-move given in (i) by reversing the orientation of the string pointed from the lower right to the upper left. We have a similar sequence for a $v\Delta_{i}^{\circ}$-move $(i=2,3,4)$ as shown in the figure. ∎

(i) It is sufficient to prove

The sequence in the top row of Figure 2.4 shows that a $v\Delta_{2}^{\wedge}$-move is realized by a combination of a $\Delta$-move, a $v\Delta_{1}^{\wedge}$-move, and a generalized Reidemeister move. Since a $\Delta$-move is realized by a combination of two crossing changes and a generalized Reidemeister move, and a crossing change is realized by a $v\Delta_{1}^{\wedge}$-move by Lemma 2.1(i), we have $v\Delta_{1}^{\wedge}\Rightarrow v\Delta_{2}^{\wedge}$. The remaining cases are proved similarly as shown in the figure, where $\stackrel{{\scriptstyle\rm cc}}{{\longleftrightarrow}}$ means a combination of crossing changes at real crossings.

(ii) It is sufficient to prove

Each of the implications can be proved by reversing the orientation of a certain string in a sequence given in (i). For example, Figure 2.5 shows $v\Delta_{1}^{\circ}\Rightarrow v\Delta_{2}^{\circ}$.

(iii) By (i) and (ii), it is sufficient to prove $v\Delta_{2}^{\wedge}\Rightarrow v\Delta_{1}^{\circ}$. Figure 2.6 shows that a $v\Delta_{1}^{\circ}$-move is realized by a combination of three crossing changes, three $v\Delta_{2}^{\wedge}$-moves, and several generalized Reidemesiter moves. Therefore we have $v\Delta_{2}^{\wedge}\Rightarrow v\Delta_{1}^{\circ}$ by Lemma 2.1(ii). ∎

Figure 2.8 shows that a $v\sharp_{2}$-move is realized by a combination of a $v\sharp_{1}$-move and several generalized Reidemeister moves. Thus we have $v\sharp_{1}\Rightarrow v\sharp_{2}$. The proof of $v\sharp_{2}\Rightarrow v\sharp_{1}$ is obtained from the above sequence by changing crossing information at every real crossing. ∎

The sequence in Figure 2.9 shows that a crossing change is realized by a combination of a $v\sharp_{1}$-move and several generalized Reidemeister moves. Therefore we have the conclusion by Lemma 2.3 ∎

$(\Rightarrow)$ By Lemmas 2.2(i) and 2.3, it is sufficient to prove $v\sharp_{1}\Rightarrow v\Delta_{1}^{\wedge}$. The sequence in Figure 2.10 shows that a $v\Delta_{1}^{\wedge}$-move is realized by a combination of a crossing change, a $v\sharp_{1}$-move, and several generalized Reidemeister moves. Therefore we have $v\sharp_{1}\Rightarrow v\Delta_{1}^{\wedge}$ by Lemma 2.4.

$(\Leftarrow)$ The sequence in Figure 2.11 shows that a $v\sharp_{1}$-move is realized by a combination of two crossing changes, a $v\Delta_{1}^{\wedge}$-move, a $v\Delta_{4}^{\wedge}$-move, and several generalized Reidemeister moves. Therefore we have $v\Delta_{j}^{\wedge}\Rightarrow v\sharp_{i}$ by Lemmas 2.1(i), 2.2(i), and 2.3. ∎

(i)$\Leftrightarrow$(ii). We have (i)$\Rightarrow$(ii) by Lemma 2.2(iii), and (ii)$\Rightarrow$(i) by definition.

(ii)$\Leftrightarrow$(iii). This follows from Lemma 2.5 immediately.

(i)$\Leftrightarrow$(iv). This has been proved in [5, Theorem 1.5]. ∎

Figure 3.2 shows that a $vp_{i}$-move $(i=2,3)$ is realized by a combination of a $vp_{1}$-move and several generalized Reidemeister moves. The other cases are proved similarly. ∎

The sequence in Figure 3.3 shows that a crossing change is realized by a combination of a $vp_{1}$-move and several generalized Reidemeister moves. Therefore we have the conclusion by Lemma 3.1. ∎

$(\Rightarrow)$ By Lemmas 2.2(ii) and 3.1, it is sufficient to prove $vp_{1}\Rightarrow v\Delta_{1}^{\circ}$. The sequence in Figure 3.4 shows that a $v\Delta_{1}^{\circ}$-move is realized by a combination of a crossing change, a $vp_{1}$-move, and several generalized Reidemeister moves. Therefore we have $vp_{1}\Rightarrow v\Delta_{1}^{\circ}$ by Lemma 3.2.

$(\Leftarrow)$ The sequence in Figure 3.5 shows that a $vp_{1}$-move is realized by a combination of two crossing changes, a $v\Delta_{1}^{\circ}$-move, a $v\Delta_{3}^{\circ}$-move, and several generalized Reidemeister moves. Therefore we have $v\Delta_{j}^{\circ}\Rightarrow vp_{i}$ by Lemmas 2.1(ii), 2.2(ii), and 3.1. ∎

This follows from Lemma 3.3 immediately. ∎

It is sufficient to prove

The sequence in the top of Figure 4.3 shows $FD_{1}\Rightarrow FD_{2}$ for $\varepsilon=+1$ and $FD_{4}\Rightarrow FD_{3}$ for $\varepsilon=-1$. We remark that two Reidemeister moves II appear in this sequence. Similarly, the sequence in the bottom of the figure shows $FD_{3}\Rightarrow FD_{1}$ for $\varepsilon=+1$ and $FD_{2}\Rightarrow FD_{4}$ for $\varepsilon=-1$. ∎

The sequence in Figure 4.4 shows that an $FD_{1}$-move is realized by a combination of a $v\Delta_{4}^{\circ}$-move, a $v\Delta_{3}^{\circ}$-move, and two Reidemeister moves II. By Lemmas 2.2(ii) and 4.1, we have the conclusion. ∎

We first consider the case $j=1$. The sequence in Figure 4.6 shows that an $F_{1}$-move is realized by a combination of two crossing changes and an $FD_{2}$-move. Note that a crossing change at a real crossing on a link diagram is described by changing the sign and orientation of the corresponding chord on a Gauss diagram. Therefore we have $v\Delta_{i}^{\circ}\Rightarrow F_{1}$ by Lemmas 2.1(ii) and 4.2 for any $i$.

Similarly, an $F_{j}$-move for $j\in\{2,\dots,6\}$ is realized by a combination of two crossing changes and an $FD_{k}$-move for some $k\in\{1,\ldots,4\}$. Thus we have the conclusion by Lemmas 2.1(ii) and 4.2. ∎

The sequence in Figure 4.8 shows that a nonself-chord oriented from $G_{i}$ to $G_{j}$ with sign $\varepsilon$ is replaced with a pair of nonself-chords one of which is oriented from $G_{1}$ to $G_{i}$ with sign $-\varepsilon$ and the other is from $G_{1}$ to $G_{j}$ with sign $\varepsilon$ by a combination of a $v\Delta_{k}^{\circ}$-move for some $k\in\{1,\dots,4\}$, a crossing change, and a Reidemeister move II. Therefore we have the conclusion by Lemmas 2.1(ii) and 2.2(ii). ∎

Let $G=\bigcup_{i=1}^{n}G_{i}$ be a Gauss diagram of $L$. Using forbidden (detour) moves and Reidemeister moves I, we can remove all the self-chords from $G$. By Lemmas 4.2 and 4.3, we may assume that $G$ satisfies the condition (i) up to $v\Delta_{i}^{\circ}$-moves and Reidemeister moves.

If there is a nonself-chord between $G_{i}$ and $G_{j}$ $(2\leq i\neq j\leq n)$, then we can replace it with a pair of chords between $G_{1}$ and $G_{i}$, and $G_{1}$ and $G_{j}$ by Lemma 4.5. Hence we may assume that $G$ satisfies the conditions (i) and (ii) up to $v\Delta_{i}^{\circ}$-moves and Reidemeister moves.

Finally, $G$ can be deformed into the one satisfying the conditions (i)–(vi) by forbidden (detour) moves, crossing changes, and Reidemeister moves II. Therefore we have the conclusion by Lemmas 2.1(ii), 4.2, and 4.3. ∎

Every pair of three chords appeared in a $v\Delta_{i}^{\circ}$-move has two adjacent endpoints with opposite signs $\varepsilon$ and $-\varepsilon$. See Figure 4.9. ∎

This follows by definition immediately. ∎

(i)$\Rightarrow$(iii). This follows from Lemma 4.6.

(iii)$\Rightarrow$(i). By Proposition 4.4, $L$ and $L^{\prime}$ are $v\Delta^{\circ}$-equivalent to

for some $a_{2},\dots,a_{n}$ and $a_{2}^{\prime},\dots,a_{n}^{\prime}\in{\mathbb{Z}}$, respectively. It follows from Lemmas 4.6 and 4.7 that

for any $i=2,\dots,n$. Since $M(a_{2},\dots,a_{n})=M(a_{2}^{\prime},\dots,a_{n}^{\prime})$ holds, $L$ is $v\Delta^{\circ}$-equivalent to $L^{\prime}$. ∎

By Lemmas 2.2(iii), 2.5, 3.3, and 4.3, we have the following.

Therefore we see that if two oriented virtual knots are $F$-equivalent, then they are $X$-equivalent for every $X\in\{v\Delta^{\wedge},\ v\Delta^{\circ},\ v\sharp,\ vp\}$. Since any two oriented virtual knots are $F$-equivalent [4, 6], they are $X$-equivalent. ∎

(i) For $X\in\{v\Delta^{\wedge},v\Delta^{\circ}\}$, if $K$ and $K^{\prime}$ are $X$-equivalent, then they are $v\Delta$-equivalent by definition. Thus we have ${\rm d}_{X}(K,K^{\prime})\geq{\rm d}_{v\Delta}(K,K^{\prime})$. Since it follows from [5, Proposition 2.6] that ${\rm d}_{v\Delta}(K,K^{\prime})\geq\frac{1}{2}|J(K)-J(K^{\prime})|$ holds, we have the inequality.

For $X=vp$, a $vp$-move contains two positive and two negative real crossings. Therefore a single $vp$-move changes the odd writhe by at most two.

(i) A $v\Delta^{\circ}$-move does not change the index of any chord except for the three chords involved in the move. See Figure 4.9 again. Therefore if $K$ and $K^{\prime}$ are related by a single $v\Delta^{\circ}$-move, then we have $\sum_{n\neq 0}|J_{n}(K)-J_{n}(K^{\prime})|\leq 3$.

For an integer $s\geq 2$, we consider a long virtual knot $T_{s}$ presented by a diagram as shown in the left of Figure 5.2, where the vertical twists consist of $2s$ positive real crossings and $2s-1$ virtual crossings. By taking the closure of the product of $m$ copies of $T_{s}$, we obtain an oriented virtual knot $K_{s}(m)$ as in the right of the figure.

As shown in the proof of [5, Theorem 2.9], the set $\{K_{s}(m)\mid s\geq 2\}$ gives an infinite family of oriented virtual knots with ${\rm u}_{v\Delta}(K_{s}(m))=m$. Since the long knot diagram of $T_{s}$ can be unknotted by a $v\Delta^{\wedge}$-move for the three real crossings around the region with the mark $*$, we have ${\rm u}_{v\Delta^{\wedge}}(K_{s}(m))=m$. ∎

For an integer $s\geq 1$, let $T_{s}$ be a long virtual knot presented by a diagram as shown in the top of Figure 5.3. Then its Gauss diagram is shown in the bottom of the figure, and has $4s+3$ chords $a_{i}$ $(i=1,2,\ldots,2s)$, $b_{j}$ $(j=1,2,3)$, and $c_{k}$ $(k=1,2,\ldots,2s)$ with signs

where $\varepsilon(\gamma)$ denotes the sign of a chord $\gamma$. Let $K_{s}(m)$ be an oriented virtual knot as the closure of the product of $m$ copies of $T_{s}$.

We can apply a $v\Delta^{\circ}$-move to the three real crossings $b_{1}$, $b_{2}$, and $b_{3}$ on the long knot diagram so that $T_{s}$ becomes unknotted. Thus we have ${\rm u}_{v\Delta^{\circ}}(K_{s}(m))\leq m$.

To prove ${\rm u}_{v\Delta^{\circ}}(K_{s}(m))\geq m$, we will calculate the $n$-writhe of $K_{s}(m)$ as follows. Since we have

it holds that

By Proposition 5.3(i), we have ${\rm u}_{v\Delta^{\circ}}(K_{s}(m))\geq\frac{1}{3}(2m+m)=m$, and hence ${\rm u}_{v\Delta^{\circ}}(K_{s}(m))=m$.

Furthermore for any $s>s^{\prime}$, since

holds, we have $K_{s}(m)\neq K_{s^{\prime}}(m)$. ∎

For an integer $s\geq 3$, let $T_{s}$ be a long virtual knot presented by a diagram as shown in the top of Figure 5.4. Then its Gauss diagram is shown in the bottom of the figure, and has $2s+6$ chords $a_{i}$ $(i=1,2,\ldots,2s)$, $b_{j}$ $(j=1,2,3,4)$, and $c_{k}$ $(k=1,2)$ with signs

Let $K_{s}(m)$ be an oriented virtual knot as the closure of the product of $m$ copies of $T_{s}$.

We can apply a $v\sharp$-move to $b_{1}$, $b_{2}$, $b_{3}$, and $b_{4}$ so that $T_{s}$ becomes unknotted. Thus we have ${\rm u}_{v\sharp}(K_{s}(m))\leq m$.

On the other hand, since we have

it holds that

This induces $J(K_{s}(m))=m+m+2m=4m$. Therefore we have ${\rm u}_{v\sharp}(K_{s}(m))\geq m$ by Proposition 5.2(ii), and hence ${\rm u}_{v\sharp}(K_{s}(m))=m$.

Furthermore for any $s>s^{\prime}$, since

holds, we have $K_{s}(m)\neq K_{s^{\prime}}(m)$. ∎

For an integer $s\geq 1$, let $T_{s}$ be a long virtual knot presented by a diagram as shown in the top of Figure 5.5. Then its Gauss diagram is shown in the bottom of the figure, and has $4s+8$ chords $a_{i}$ $(i=1,2,\dots,2s)$, $b_{j}$ $(j=1,2,3,4)$, $c_{k}$ $(k=1,2)$, and $d_{\ell}$ $(\ell=1,2,\dots,2s+2)$ with signs

Let $K_{s}(m)$ be an oriented virtual knot as the closure of the product of $m$ copies of $T_{s}$.

We can apply a $vp$-move to $b_{1}$, $b_{2}$, $b_{3}$, and $b_{4}$ so that $T_{s}$ becomes unknotted. Thus we have ${\rm u}_{vp}(K_{s}(m))\leq m$.

On the other hand, since we have