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Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

Department of Information and Communication Sciences, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan

Department of Mathematics, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma 371-8530, Japan

E-mail Address: shimizu@nat.gunma-ct.ac.jp

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Yoshiro Yaguchi

Department of Mathematics, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma 371-8530, Japan

E-mail Address: yaguchi-y@nat.gunma-ct.ac.jp

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https://doi.org/10.1142/S0218216517500730Cited by:1 (Source: Crossref)

Abstract

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two $2$ $2$ -component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram $D$ $D$ , there exists a diagram $D^{'}$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

Keywords:

AMSC: 57M25

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