The Second Reidemeister Moves and Colorings of Virtual Knot Diagrams

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.