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Local Moves Generating Writhe Polynomials of Virtual Knots

Takuji Nakamura (Osaka Electro‐Communication University)
Yasutaka Nakanishi (Kobe University)
Shin Satoh (Kobe University)
1

Introduction

This paper is a résumé of the paper [8]. The odd writhe J(K) is one of fundamental
invariants of a virtual knot K (cf. [5]). In [10], Taniguchi and the third author intro‐
duced local moves called :‐moves and proved that :‐moves correspond to the odd writhe;
namely, two virtual knots have the same odd writhe if and only if they are related by afinite sequence of 三‐moves.
On the other hand, there is another invariant of a virtual knot K called the writhe

polynomial W_{K}(t) (cf. [1, 6, 10]). The writhe polynomial is stronger than the odd writhein the sense that J(K)=-W_{K}(-1)/2 holds. Therefore, it is natural to ask which local
moves correspond to the writhe polynomial.

In Section 2, we introduce local moves for virtual knots and links called shell moves. InSection 3, we prove that the writhe polynomial correspond to shell moves. In Section 4,
we study which invariants of a 2‐component virtual link correspond to shell moves.

2Gauss diagrams

A Gauss diagram consists of a finite number of oriented circles and oriented, signed chordsequipped with signs of endpoints as shown in Figure 1.
\prime.<'\cdots\cdot\prime^{\prime'}. \prec\cdots\cdot\prime\prec-\ dot\cdot\cdot\cdot\backsla h\backsla h\backsla h\dot{j}Figure 1: Self‐ and nonself‐chords of a Gauss diagram

A \mu ‐component virtual link is an equivalence class of Gauss diagrams with \mu circles up

to Reidemeister moves RI‐R3 (cf. [4, 5]). See Figure 2. If \mu=1 , it is called a virtualknot.

The first author is partially supported by JSPS Grants‐in‐Aid for Scientific Research (C), 17K05265 . The second author
is partially supported by JSPS Grants‐in‐Aid for Scientific Research (C), 19K03492 . The third author is partially supportedby JSPS Grants‐in‐Aid for Scientific Research (C), 19K03466.

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R2R2\sim\sim\sim R3Figure 2: Reidemeister moves for Gauss diagrams

The shells for a chord \gamma are parallel self‐chords which are oriented with respect to thesign of the endpoint of \gamma as shown in Figure 3. We remark that a shell in this paper is a

special case of an anklet in [7].

Figure 3: Shells for \gamma

The shell moves Sl and S2 are local moves defined by using Gauss diagrams as shownin Figure 4. Precisely, an Sl‐move slides a shell along a chord to the opposite side withthe same sign, and an S2‐move changes the position of the adjacent endpoints of chords
with making a pair of shells with respect to the signs of the endpoints.
S1

\sim

Figure 4: Shell moves Sl and S2

We say that two Gauss diagram are S‐equivalent if they are related by a finite sequenceof Reidemeister moves and shell moves, and two virtual links are S‐equivalent if their Gauss
diagrams are S ‐equivalent.

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Lemma 2.1. If two Gauss diagrams are related by a deformation (1) or (2) as shown in
Figure 5, then they are S‐equivalent.
the suofsignsy\square

(1)
\sim

\sim_{-\delta_{1}^{-} (2)Figure 5:\Sigma^{\delta_{J}\delta_{j}S ‐equivalent Gauss diagrams in Lemma 2.1

Lemma 2.2. If two Gauss diagrams are related by a deformation (1) -(4) as shown in
Figure 6, then they are S‐equivalent. Here,

Pand Q are portions of whole chords.

\square(1)\sim\sim(2)
(3)
'(4)
\sim

Figure 6:

S ‐equivalent Gauss diagrams in Lemma 2.2

Let G be a Gauss diagram with \mu circles C_{1} , .
C_{\mu} . For n\in \mathbb{Z} and 1\leq i\neq j\leq\mu,we define the n ‐snail of type i and the n ‐snail of type (i, j) to be the portion of chords asshown in Figure 7. Then we have the following.

Lemma 2.3. If two Gauss diagrams are related by a deformation (1) -(4) as shown inFigure 8, then they are S‐equivalent.

\squareBy using Lemmas 2.1−2.3, we have the following standard form of a Gauss diagram up
to S ‐moves.

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um0\dagger=-\epsilon n\in S_{i}(n) \epsilon 5_{ij}(n)Figure 7: The n ‐snails \varepsilon S_{\dot{i} (n) and \varepsilon S_{ij}(n)(1)\simc_{i}arrow(2)\sim(3)C_{j}arrow\sim

C_{i}arrow
(4)
\sim

\subset_{i}arrow

Figure 8:

S ‐equivalent Gauss diagrams in Lemma 2.3

Proposition 2.4. Any Gauss diagram of an oriented \mu ‐component virtual link is S‐
equivalent to a Gauss diagram G with \mu circles C_{1} , .
C_{\mu} which satisfies the following
conditions. Figure 9 shows the case \mu=3.

(i) The chords of G form a finite number of snails.
(ii) There is an arc \alpha_{i} on each C_{i} such that all snails of type i spans \alpha_{i}.
(iii) All snails of type (i, j) spans (C_{i}\backslash \alpha_{i})\cup(C_{j}\backslash \alpha_{j}) in parallel.
(iv) There is no snails \pm S_{i}(0) or\pm S_{i}(1) for any i.

(v) There is no pair of snails +S_{i}(n) and-S_{\dot{i}}(n) for any i and n.
(vi) There is no pair of snails +S_{\dot{\iota}j}(n) and-S_{ij}(n) for any i\neq j and n.

3

The case\mu=1

In this section, we consider an oriented virtual knotcircle

C.

\squareFor portions P_{1} , .

K and its Gauss diagram G with aP_{k} of chords, we denote by( \sum_{i=1}^{k}P_{i}) the Gauss diagram

as shown in Figure 10. For integers a, n\in \mathbb{Z}, aS(n) denotes the concatenation of |a|copies of \varepsilon S(n) , where \varepsilon is the sign of a.

Lemma 3.1. Any Gauss daigram of K isa_{n}\in \mathbb{Z}.

S ‐equivalent to( \sum_{n\neq 0,1}a_{n}S(n) for some\square

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Figure 9: A Gauss diagram with three circles

Figure 10: The Gauss diagram( \sum_{i=1}^{k}P_{i})A chord \gamma divides the circle C into two arcs. Let \alpha be the one oriented from the initial

to the terminal endpoint of \gamma . We define the index of \gamma to be the sum of signs of endpointsof chords on \alpha . For each n\neq 0 , the sum of signs of all chords whose index is equal ton defines an invariant of K . It is called the n ‐writhe of K and denoted by
J_{n}(K) . The
writhe polynomial is defined by

W_{K}(t)= \sum_{n\neq 0}J_{n}(K)(t^{n}-1)\in \mathbb{Z}[t, t^{-1}].
Refer to [1, 2, 6, 10] for more details.
Lemma 3.2. Let K be an oriented virtual knot.

(i) The writhe polynomial W_{K}(t) is invariant under S‐moves.
(ii) If K is presented by a Gauss diagram given in Lemma 3.1, then we have

W_{K}(t)= \sum_{n\neq 0,1}a_{n}t^{n}-(\sum_{n\neq 0,1}na_{n})t+\sum_{n\neq 0,1} (n-1)a_{n}.
\squareBy Lemma 3.1 and Lemma 3.2(ii), we have the following.
Proposition 3.3. Let K and K' be oriented virtual knots. If W_{K}(t)=W_{K'}(t) holds,\squarethen K and K' are S‐equivalent.

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Therefore the following holds by Lemma 3.2(i) and Proposition 3.3.
Theorem 3.4. For two oriented virtual knots K and K' , the following are equivalent.

(i) W_{K}(t)=W_{K'}(t) .
and K' are related by a finite sequence of shell moves.

(ii)

K4

The case\square\mu=2

In this section, we consider an oriented 2‐component virtual link L=K_{1}\cup K_{2} and its
Gauss diagram G with a pair of circles C_{1} and C_{2} . By Proposition 2.4, we have thefollowing.

Lemma 4.1. Any Gauss daigram of L is S ‐equivalent to a Gauss diagram( \sum_{n\neq 0,1}a_{n}S_{1}(n), \sum_{n\neq 0,1}b_{n}S_{2}(n);\sum_{m\in Z} c_{m}S_{12}(m), \sum_{m\in Z}d_{m}S_{21}(m)for some integers a_{n}, b_{n}(n\neq 0,1) and c_{m}, d_{m}(m\in \mathbb{Z}) as shown in Figure 11. Here, theentries present the concatenations of snails of type 1, 2, (1, 2), and (2, 1), respectively. \square

Figure 11: A Gauss diagram of an oriented 2‐component virtual link

For (i, j)=(1,2) or (2, 1), the (i, j) ‐linking number of L , denoted by Lk(K_{i}, K_{j}) , isdefined to be the sum of signs of all nonself‐chords oriented from C_{i} to C_{j} . The virtual
linking number of L is defined by \lambda(L)=Lk(K_{1}, K_{2})-Lk(K_{2}, K_{1}) [9] (cf. [3]). It is easyto see that Lk(K_{1}, K_{2}), Lk(K_{2}, K_{1}) , and \lambda(L) are invariant under S ‐moves.
If \lambda(L)<0 , then by switching the roles of K_{1} and K_{2} , the case reduces to \lambda(L)>0.
In what follows, we may assume that \lambda(L)\geq 0 . We denote \lambda(L) by \lambda for simplicity. The
following propositions give standard forms of L up to S ‐equivalence.
Proposition 4.2. Let G be a Gauss diagram of L.

(i) If \lambda\geq 1 , thenG \sim(\sum_{n\neq 0,1-\lambda,-\lambda+1}a_{n}S_{1}(n),\sum_{n\neq 0,1 \lambda,\lambda+1}b_{n}S_{2}(n)

;

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\sum_{m=0}^{\lambda-1}c_{m}S_{12}(p+m),\sum_{m=0}^{\lambda-1}d_{m}S_{21}(-p-m) )for some integers a_{n}(n\neq 0,1, -\lambda, -\lambda+1), b_{n}(n\neq 0,1, \lambda, \lambda+1), c_{m}, d_{m}(0\leq m\leq\lambda-1) , and p.

(ii) In particular, if \lambda=1 , thenG \sim(\sum_{n\neq 0,1,-1}a_{n}S_{1}(n), \sum_{n\neq 0,12},b_{n}S_{2}(n);c_{0} S_{12}(0), d_{0}S_{21}(0)for some integers a_{n}(n\neq 0,1, -1), b_{n}(n\neq 0,1,2), c_{0} , and d_{0}.

\squareProposition 4.3. We have the following S‐equivalent Gauss diagrams.

(i) If \lambda=0 , then(P, Q; \sum_{m\in \mathb {Z} c_{m}S_{12}(m),\sum_{m\in \mathb {Z} d_{m}S_{21} (m)\sim(P, Q;\sum_{m\in \mathb {Z} c_{m}S_{12}(m+k),\sum_{m\in \mathb {Z} d_{m}S_ {21}(m-k)

for any k\in \mathbb{Z}.

(ii) If \lambda\geq 2 , then(P, Q; \sum_{m=0}^{\lambda-1}c_{m}S_{12}(p+m),\sum_{m=0}^{\lambda-1}d_{m}S_{21} (-p-m)\sim(P, Q;\sum_{m=0}^{\lambda-1}c_{m}'S_{12}(p'+m),\sum_{m=0}^{\lambda-1}d_{m} 'S_{21}(-p'-m)

,

where

\{ begin{ar ay}{l} (c\'{O},. c_{\lambda-k1}',c_{\lambda-k}',\ldots,c_{\lambda-1}')=(c_{k}, \ldots,c_{\lambda-1},c_{0},\ldots,c_{k-1}), (d\'{O} d_{\lambda-k1}',d_{\lambda-k}',\ldots,d_{\lambda-1}')=(d_{k}, \ldots,d_{\lambda-1},d_{0},\ldots,d_{k-1}), \end{ar ay}
and p'=p+k- \sum_{i=0}^{k-1}(c_{i}-d_{i}) for any k with 1\leq k\leq\lambda-1.

\squareA self‐chord \gamma spanning C_{i} divides C_{i} into two arcs. Let \alpha be the one oriented from theinitial to the terminal endpoint of \gamma in C_{i} . we define the index of \gamma in G to be the sumof signs of endpoints of self‐ and nonself‐chords on \alpha . For n\in \mathbb{Z} and i=1,2 , the sumof signs of all self‐chords whose indices are equal to n defines the invariants J_{n}(K_{1};L) forn\neq 0, -\lambda and J_{n}(K_{2};L) for n\neq 0, \lambda . These are called the n ‐writhes of K_{1} and K_{2} in L,
respectively.

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On the other hand, for nonself‐chords \gamma and \gamma_{0} , the relative index of \gamma with respect to\gamma_{0} to be the index of \gamma in the Gauss diagram obtained from G by surgery along \gamma_{0} . Forn\in \mathbb{Z} and (i, j)=(1,2), (2,1) , let J_{n}^{ij}(G;\gamma_{0}) denote the sum of signs of nonself‐chords \gamma
oriented from C_{i} to C_{j} whose relative index with respect to \gamma_{0} are equal to n . Put

F_{ij}(t; \gamma_{0})=\sum_{n\in \mathb {Z} J_{n}^{ij}(G;\gamma_{0})t^{n}.
For an integer s\geq 0 , let \Lambda_{s} denote the Laurent polynomial ring \mathbb{Z}[t, t^{-1}]/(t^{s}-1) . In
particular, we have \Lambda_{0}=\mathbb{Z}[t, t^{-1}] and \Lambda_{1}=\mathbb{Z} . We consider an equivalence relation on\Lambda_{s}\cross\Lambda_{s} such that (f_{1}(t), g_{1}(t)) and (f_{2}(t), g_{2}(t)) are equivalent if there is an integer k withf_{2}(t)=t^{k}f_{1}(t) and g_{2}(t)=t^{-k}g_{1}(t) .
We denote by [f(t), g(t)] the equivalence class represented by (f(t), g(t)) , and by \Gamma(s) theset of such equivalence classes.
Then the equivalence class [F_{12}(t;\gamma_{0}), F_{21}(t;\gamma_{0})]\in\Gamma(\lambda) defines the invariant of L

(cf. [2]). We call it the linking class of L and denote it by F(L) . In particular, F(L)=(Lk(K_{1}, K_{2}), Lk(K_{2}, K_{1}))\in \mathbb{Z}\cross \mathbb{Z} for \lambda=1 . Then by Propositions 4.2 and 4.3, we havethe following.
Theorem 4.4. Let L=K_{1}\cup K_{2} and L'=K_{1}'\cup K_{2}' be oriented 2‐component virtual links
with \lambda=\lambda'=0 . Then L and L' are related by a finite sequence of shell moves if and only
if(i) J_{n}(K_{1};L)=J_{n}(K_{1}';L') for any n\neq 0,1,(ii) J_{n}(K_{2};L)=J_{n}(K_{2}'; L') for any n\neq 0,1 , and(iii) F(L)=F(L') .
\squareTheorem 4.5. Let L=K_{1}\cup K_{2} and L'=K_{1}'\cup K_{2}' be oriented 2‐component virtual links
with \lambda=\lambda'=1 . Then L and L' are related by a finite sequence of shell moves if and only
if(i) J_{n}(K_{1};L)=J_{n}(K_{1}';L') for any n\neq 0,1, -1,(ii) J_{n}(K_{2};L)=J_{n}(K_{2}'; L') for any n\neq 0,1,2 , and(iii) F(L)=F(L') .
\squareTheorem 4.6. Let L=K_{1}\cup K_{2} and L'=K_{1}'\cup K_{2}' be oriented 2‐component virtual links
with \lambda=\lambda'\geq 2 . Then L and L' are related by a finite sequence of shell moves if and onlyif115(i) J_{n}(K_{1};L)=J_{n}(K_{1}';L') for any n\neq 0,1, -\lambda, -\lambda+1,(ii) J_{n}(K_{2};L)=J_{n}(K_{2}'; L') for any n\neq 0,1, \lambda, \lambda+1,(iii) F(L)=F(L') , and(iv) J_{1}(K_{1};L)+J_{-\lambda+1}(K_{1};L)+J_{1}(K_{2};L)+J_{\lambda+1}(K_{2};L)=J_{1} (Kí; L' ) +J_{-\lambda+1} (Kí; L' ) +J_{1}(K_{2}';L')+J_{\lambda+1}(K_{2}';L') .
\squareReferences[1] Z. Cheng, A polynomial invariant of virtual knots, Proc. Amer. Math. Soc. 142(2014), no. 2, 713‐725.
[2] Z. Cheng and H. Gao, A polynomial invariant of virtual links, J. Knot Theory Ram‐ifications 22 (2013), no. 12, 1341002, 33 pp.
[3] L. C. Folwaczny and L. H. Kauffman, A linking number definition of the affine in‐dex polynomial and applications, J. Knot Theory Ramifications 22 (2013), no. 12,1341004, 30 pp.

[4] M. Goussarov, M. Polyak, and O. Viro, Finite‐type invariants of classical and virtualknots, Topology 39 (2000), no. 5, 1045‐1068.

[5] L. H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663‐690.
[6] L. H. Kauffman, An affine index polynomial invariant of virtual knots, J. Knot Theory
Ramifications 22 (2013), no. 4, 1340007, 30 pp.
[7] T. Nakamura, Y. Nakanishi, and S. Satoh, A note on coverings of virtual knots,available at arXiv: 1811. 10852[8] T. Nakamura, Y. Nakanishi, and S. Satoh, Writhe polynomials and shell moves forvirtual knots and links, in submission.

[9] T. Okabayashi, Forbidden moves for virtual links, Kobe J. Math. 22 (2005), no. 1‐2,49‐63.

[10] S. Satoh and K. Taniguchi, The writhes of a virtual knot, Fund. Math. 225 (2014),
no. 1, 327‐342.

Department of Engineering Science
Osaka Electro‐Communication University
Osaka 572‐8530

JAPANE‐mail address:n‐takuji@osakac.ac.jp

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Department of Mathematics
Kobe University
Kobe 657‐8501

JAPANE‐mail address: nakanisi@math.kobe‐u.ac.jp, shin@math.kobe‐u.ac.jp