Graduate School of Natural Sciences, Nagoya City University
1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, Aichi 467-8501 Japan
This work was supported by JSPS KAKENHI Grant Number 15K04879.
###### Abstract

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

## 1 Introduction

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.

## 2 The JKSS invariant

We recall the definition of the JKSS invariant of a virtual link. Let be a virtual link diagram with real crossings. Let be the real crossings of . We define a matrix, , where (or ) if is positive (or negative) crossing. Here and are matrices: and . Let be the 4-valent graph obtained from by regarding each real crossing as a vertex of . We denote by the same symbols the vertices of . The graph is immersed in and the multiple points of are virtual crossings of . For each vertex of , consider an open regular neighborhood of in . Then is the union of four open arcs, which we call the short edges around . According to the position, we denote by the short edges as in Figure 5.

We define a matrix, as follows. For each ,

 p(2i−1+ϵ)(2j−1+λ)={1(\parbox284.527559ptiftwoshortedges$i−ϵ$and$j+λ$areonthesameedgeof$|D|$)0\parbox85.358268pt(otherwise),

where .

###### Theorem 1 (Jaeger, Kauffman and Saleur [3], Sawollek [7])

For a virtual link diagram , is an invariant of the virtual link up to multiplication by powers of , i.e., for any virtual link diagram representing the same virtual link with , we have for some .

The JKSS invariant of a virtual link is defined by for a diagram of . For example, the JKSS invariant of a virtual link depicted in Figure 6 is .

We define an invariant of twisted links which is related to the JKSS invariant. Let be a twisted link diagram with real crossings .

We define a matrix , by , where (or ) if the crossing is positive (or negative). Here and are matrices and respectively, i.e.,

 ˜M+=⎛⎜ ⎜ ⎜ ⎜⎝1−x−y00−xy−1000001−x−y00−xy−10⎞⎟ ⎟ ⎟ ⎟⎠ and ˜M−=⎛⎜ ⎜ ⎜ ⎜⎝0−x−1y00−y−11−x−100000−x−1y00−y−11−x−1⎞⎟ ⎟ ⎟ ⎟⎠.

For a twisted link diagram , the graph is defined by the same way before. Each edge of may have bars on it. For each vertex of , we denote by , and the short edge around as before. We denote (or ), if two short edges and are on the same edge of and there are an even (or odd) number of bars on the edge.

We defined matrix, as follows. For each ,

 ~p(4i−3+a)(4j−3+b)=⎧⎨⎩1(\parbox256.074803pt$iae←jb$or$i3−ae←j3−b$or$iao←j3−b$or$i3−ao←jb$)0\parbox85.358268pt(otherwise),

where Note that and arc not defined for . We assume that and are false when or .

###### Theorem 2

For a twisted link diagram , is an invariant of the twisted link up to multiplication by powers of , i.e., for any twisted link diagram representing the same twisted link with , we have for some .

For a twisted link , we define the twisted JKSS invariant of , denoted by , by for a diagram of .

The twisted JKSS invariant of the diagram in Figure 7 (a) is and that of the diagram in Figure 7 (b) is . 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 .

## 3 Proof of Theorem 2

Let be a twisted link diagram with bars . Assume that is on the left of the -axis and all bars are parallel to the -axis with disjoint -coordinates. Let be the twisted link diagram which is obtained from by reflection with respect to the -axis and switching all real crossings of . See Figure 8.

We construct the double covering of as follows:

For horizontal lines such that contains and , we replace each part of in a neighborhood of as in Figure 9. We call this diagram the double covering diagram of .

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

###### Theorem 3 ([5])

Let and be twisted link diagrams and and their double coverings diagrams of and . If and are equivalent as twisted links, then and are equivalent as virtual links.

###### Theorem 4

For a twisted link diagram , coincides to , where is the double covering of .

Proof Let be a twisted link diagram with real crossings . Let be the real crossings of corresponding to . We regard as the real crossings of the double covering diagram . We rename the real crossings of by such that and for . Then the matrix for coincides with matrix for . We show that the matrix for coincides with the matrix for . Suppose that real crossings and of are the boundary points of an edge of , i.e., or holds for .

1. If for , we see that and (or and ) are the boundary points of an edge of . The short edges labeled by and (or and ) are on the same edge of . Thus we have

 ~p(4i−3+a)(4j−3+b)=⎧⎪⎨⎪⎩1(a=ϵ and% b=λ)1(a=2+1−ϵ and b=2+1−λ)0(otherwise).
2. If for , we see that and (or and ) are the boundary points of an edge of . Namely, the short edges labeled by and (or and ) are on the same edge of . Thus we have

 ~p(4i−3+a)(4j−3+b)=⎧⎪⎨⎪⎩1(a=ϵ and% b=2+1−λ)1(a=2+1−ϵ and b=λ)0(otherwise).

Thus we have the . Since is , .

## 4 Properties of double covering diagrams

For a twisted (virtual or classical) link diagram , we denote by the number of real crossings of . For a twisted link (or a virtual link) , we denote by (or ) the minimal number of real crossings of all of diagrams of .

###### Theorem 5

Let be a twisted link presented by a twisted link diagram and let be the virtual link presented by the double covering diagram of . If , then .

Proof Note that . If , then there is a diagram of , such that . For the double covering diagram of , we have , which is a contradiction of the assumption since is equivalent to as a virtual link.

###### Proposition 6

Let be a twisted link diagram obtained from a virtual link diagram by adding a bar on an arc. Then is equivalent to a connected sum of and modulo virtual Reidemeister moves.

Proof For the double covering of , there are two kinds of sets of virtual crossings. One is the set of virtual crossings which correspond to virtual crossings of and , and the other is the set of virtual crossings which occur when we construct from by replacement as in Figure 9. The second set of virtual crossings of look as in Figure 11 (i). As in Figure 11 (ii), such virtual crossings are eliminated by some virtual Reidemeister moves. The virtual link diagram shown in Figure 11 (ii) is a connected sum of two virtual link diagrams which are equivalent to and modulo virtual Reidemeister moves.

Acknowledgements

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

## References

• [1] M. O. Bourgoin, Twisted link theory, Algebr. Geom. Topol. 8 (2008) 1249–1279.
• [2] J. S. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002) 311–322.
• [3] F. Jaeger, L. H. Kauffman, H. Saleur, The Conway polynomial in and in thickened surfaces: a new determinant formulation, Combin. Theory Ser. B 61 (1994), 237–259.
• [4] N. Kamada and S. Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9 (2000) 93-106.
• [6] L. H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663–690.
• [7] J. Sawollek, On Alexander-Conway polynomials for virtual knots and links, preprint (1999, arXiv:9912173(math.GT)).
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