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Clasp-pass moves and Kauffman polynomials of virtual knots

Myeong-Ju Jeong

Department of Mathematics, Korea Science Academy of KAIST, Busan 614-822, Republic of Korea

E-mail Address: mjjeong@kaist.ac.kr

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and

Dahn-Goon Kim

Department of Mathematics, Korea Science Academy of KAIST, Busan 614-822, Republic of Korea

E-mail Address: morningking@naver.com

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https://doi.org/10.1142/S0218216516500450Cited by:0 (Source: Crossref)

Abstract

Habiro showed that two knots $K_{1}$ $K_{1}$ and $K_{2}$ $K_{2}$ are related by a finite sequence of clasp-pass moves, if and only if they have the same value for Vassiliev invariants of type $< 3$ $< 3$ . Tsukamoto showed that, if two knots differ by a clasp-pass move then the values of the Vassiliev invariant $V_{K}^{‴} (1)$ of degree $3$ for the two knots differ by $\pm 36$ or $0$ , where $V_{K} (t)$ is the Jones polynomial of a knot $K$ . If two virtual knots are related by clasp-pass moves, then they take the same value for all Vassiliev invariants of degree $< 3$ . We extend the Tsukamoto’s result to virtual knots by using a Vassiliev invariant $v_{3}$ of degree $3$ , which is induced from the Kauffman polynomial. We also get a lower bound for the minimal number of clasp-pass moves needed to transform $K_{1}$ to $K_{2}$ , if two virtual knots $K_{1}$ and $K_{2}$ can be related by a finite sequence of clasp-pass moves.

Keywords:

AMSC: 57M25, 57M27

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