A twisted link invariant derived from a virtual link invariant

L. H. Kauffman introduced virtual knot theory, which is a generalization of knot theory based on Gauss chord diagrams and link diagrams in closed oriented surfaces [6]. Twisted knot theory was introduced by Bourgoin. It is an extension of virtual knot theory. Twisted links correspond to stable equivalence classes of links in oriented 3-manifolds which are line bundles over (possibly non-orientable) closed surfaces [1], [2].

F. Jaeger, L. H. Kauffman and H. Saleur defined an invariant of links in thickened surfaces, where the surfaces are oriented [3]. J. Sawollek [7] applied it to virtual links, which is so-called the JKSS invariant. In this paper, we introduce an invariant of twisted links obtained from the JKSS invariant by use of double coverings. We also discuss some properties of double coverings of twisted link diagrams.

A virtual link diagram is a generically immersed loops whose double points have information of positive, negative or virtual crossing. A virtual crossing is an encircled double point without over-under information. A twisted link diagram is a virtual link diagram, possibly with bars on arcs. Examples of twisted link diagrams are depicted in Figure 1.

A twisted link (resp. a virtual link) is an equivalence class of a twisted link diagram (resp. virtual link diagram) under Reidemeister moves, virtual Reidemeister moves and twisted Reidemeister moves (resp. Reidemeister moves and virtual Reidemeister moves) depicted in Figures 2, 3 and 4.

We recall the definition of the JKSS invariant of a virtual link. Let $D$ be a virtual link diagram with $n$ real crossings. Let $c_1,\dots ,c_n$ be the real crossings of $D$. We define a $2n\times 2n$ matrix, $M=diag(M_1,\dots ,M_n)$, where $M_i=M_{+}$ (or $M_{-}$) if $c_i$ is positive (or negative) crossing. Here $M_{+}$ and $M_{-}$ are $2\times 2$ matrices: $M_{+}=\left(\begin{array}{ccc}1-x& -y& \\ -x{y}^{-1}& 0& \end{array}\right)$ and $M_{-}=\left(\begin{array}{cc}0& -x^{-1}y\\ -y^{-1}& 1-x^{-1}\end{array}\right)$. Let $|D|$ be the 4-valent graph obtained from $D$ by regarding each real crossing as a vertex of $|D|$. We denote by the same symbols $c_1,\dots ,c_n$ the vertices of $|D|$. The graph $|D|$ is immersed in $R^2$ and the multiple points of $|D|$ are virtual crossings of $D$. For each vertex $c_i$ of $|D|$, consider an open regular neighborhood $N(c_i,|D|)$ of $c_i$ in $|D|$. Then $N(c_i,|D|)-\left\{c_i\right\}$ is the union of four open arcs, which we call the short edges around $c_i$. According to the position, we denote by $i_0^{-},i_1^{-},i_0^{+},i_1^{+}$ the short edges as in Figure 5.

We define a $2n\times 2n$ matrix, $P=\left(p_{kl}\right)$ as follows. For each $i,j\in \{1,\dots ,n\}$,

where $ϵ,\lambda \in \{0,1\}$.

The JKSS invariant of a virtual link $L$ is defined by $Z_L(x,y)=Z_D(x,y)$ for a diagram $D$ of $L$. For example, the JKSS invariant of a virtual link depicted in Figure 6 is $(x-1)(y+1)(x+y)y^{-1}$.

We define an invariant of twisted links which is related to the JKSS invariant. Let $D$ be a twisted link diagram with $n$ real crossings $c_1,\dots ,c_n$.

We define a $4n\times 4n$ matrix $\tilde{M}$, by $\tilde{M}=diag({\tilde{M}}_1,\dots ,{\tilde{M}}_n)$, where ${\tilde{M}}_i={\tilde{M}}_{+}$ (or ${\tilde{M}}_{-}$) if the crossing $c_i$ is positive (or negative). Here ${\tilde{M}}_{+}$ and ${\tilde{M}}_{-}$ are $4\times 4$ matrices $diag(M_{+},M_{+})$ and $diag(M_{-},M_{-})$ respectively, i.e.,

For a twisted link diagram $D$, the graph $|D|$ is defined by the same way before. Each edge of $|D|$ may have bars on it. For each vertex $c_i$ of $|D|$, we denote by $i_0^{-},i_1^{-},i_0^{+}$, and $i_1^{+}$ the short edge around $c_i$ as before. We denote $i_{ϵ}\stackrel{e}{\leftarrow }{j}_{\lambda }$ (or $i_{ϵ}\stackrel{o}{\leftarrow }{j}_{\lambda }$), if two short edges $i_{ϵ}^{-}$ and $j_{\lambda }^{+}$ are on the same edge of $|D|$ and there are an even (or odd) number of bars on the edge.

We defined $4n\times 4n$ matrix, $\tilde{P}=\left({\tilde{p}}_{kl}\right)$ as follows. For each $i,j\in \{1,\dots ,n\}$,

where $a,b\in \{0,1,2,3\}$ Note that $i_k^{-}$ and $j_k^{-}$ arc not defined for $k\in \{2,3\}$. We assume that $i_k\stackrel{e}{\leftarrow }{j}_l$ and $i_k\stackrel{o}{\leftarrow }{j}_l$ are false when $k\in \{2,3\}$ or $l\in \{2,3\}$.

For a twisted link $L$, we define the twisted JKSS invariant of $L$, denoted by ${\tilde{Z}}_L(x,y)$, by ${\tilde{Z}}_D(x,y)$ for a diagram $D$ of $L$.

The twisted JKSS invariant of the diagram in Figure 7 (a) is $0$ and that of the diagram in Figure 7 (b) is $y^{-2}\left(x^2-1\right)\left(y^2-1\right)\left(x^2-y^2\right)$. We see that they are not equivalent. Note that they are not distinguished by the twisted Jones polynomial defined in [1]. The twisted Jones polynomials of the twisted links in Figure 7 are $-A^{-6}(A^4+A^{-4})M$.

Let $D$ be a twisted link diagram with bars $b_1,\dots ,b_k$. Assume that $D$ is on the left of the $y$-axis and all bars are parallel to the $x$-axis with disjoint $y$-coordinates. Let $s\left(D\right)$ be the twisted link diagram which is obtained from $D$ by reflection with respect to the $y$-axis and switching all real crossings of $D$. See Figure 8.

We construct the double covering of $D$ as follows:

For horizontal lines $l_1,\dots ,l_k$ such that $l_i$ contains $b_i$ and $s\left(b_i\right)$, we replace each part of $D⨿s\left(D\right)$ in a neighborhood of $N\left(l_i\right)$ as in Figure 9. We call this diagram $\tilde{D}$ the double covering diagram of $D$.

The double covering diagram of the twisted link diagram $D$ in Figure 8 is shown in Figure 10.

1. If $i_{ϵ}\stackrel{e}{\leftarrow }{j}_{\lambda }$ for $D$, we see that $c_i$ and $c_j$ (or $s\left(c_i\right)$ and $s\left(c_j\right)$ ) are the boundary points of an edge of $|\tilde{D}|$. The short edges labeled by $(2i-1{)}_{ϵ}^{-}$ and $(2j-1{)}_{\lambda }^{+}$ (or $(2i{)}_{1-ϵ}^{-}$ and $(2j{)}_{1-\lambda }^{+}$) are on the same edge of $|\tilde{D}|$. Thus we have

   $${\tilde{p}}_{(4i-3+a)(4j-3+b)}=\{\begin{array}{cc}1& (a=ϵ\text{and\%}b=\lambda )\\ 1& (a=2+1-ϵ\text{and}b=2+1-\lambda )\\ 0& \left(\text{otherwise}\right).\end{array}$$

2. If $i_{ϵ}\stackrel{o}{\leftarrow }{j}_{\lambda }$ for $D$, we see that $c_i$ and $s\left(c_j\right)$ (or $s\left(c_i\right)$ and $c_j$ ) are the boundary points of an edge of $|\tilde{D}|$. Namely, the short edges labeled by $(2i-1{)}_{ϵ}^{-}$ and $(2j{)}_{1-\lambda }^{+}$ (or $(2i{)}_{1-ϵ}^{-}$ and $(2j-1{)}_{\lambda }^{+}$) are on the same edge of $|\tilde{D}|$. Thus we have

   $${\tilde{p}}_{(4i-3+a)(4j-3+b)}=\{\begin{array}{cc}1& (a=ϵ\text{and\%}b=2+1-\lambda )\\ 1& (a=2+1-ϵ\text{and}b=\lambda )\\ 0& \left(\text{otherwise}\right).\end{array}$$

Thus we have the $\tilde{P}=P$. Since $w\left(\tilde{D}\right)$ is $2w\left(D\right)$, $Z_{\tilde{D}}(x,y)=det(M-P)=det(\tilde{M}-\tilde{}P)={\tilde{Z}}_D(x,y)$. $\square$

For a twisted (virtual or classical) link diagram $D$, we denote by $r\left(D\right)$ the number of real crossings of $D$. For a twisted link (or a virtual link) $L$, we denote by $r\left(L\right)$ (or $r_v\left(L\right)$) the minimal number of real crossings of all of diagrams of $L$.

Acknowledgements

The author would like to thank Seiichi Kamada for his useful suggestion.