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OPEN ACCESS eISSN 0454-8124
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Kyungpook Mathematical Journal 2022; 62(2): 347-361

Published online June 30, 2022

Copyright © Kyungpook Mathematical Journal.

The Second Reidemeister Moves and Colorings of Virtual Knot Diagrams

Myeong–Ju Jeong*, Yunjae Kim

Department of Mathematics, Korea Science Academy, 111 Baekyang Gwanmun–Ro, Busanjin–Gu, Busan 47162 Korea
e-mail : mjjeong@ksa.kaist.ac.kr

Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566 Korea
e-mail : kimholzi@gmail.com

Received: April 22, 2020; Revised: February 10, 2022; Accepted: February 22, 2022

Abstract

Two virtual knot diagrams are said to be equivalent, if there is a sequence S of Reidemeister moves and virtual moves relating them. The difference of writhes of the two virtual knot diagrams gives a lower bound for the number of the first Reidemeister moves in S. In previous work, we introduced a polynomial qK(t) for a virtual knot diagram K which gave a lower bound for the number of the third Reidemeister moves in the sequence S. In this paper we define a new polynomial from a coloring of a virtual knot diagram. Using this polynomial, we give a lower bound for the number of the second Reidemeister moves in S. The polynomial also suggests the design of the sequence S.

Keywords: virtual knot, Reidemeister moves, coloring, knot polynomial