Reidemeister moves and parity polynomials of virtual knot diagrams

Abstract

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial qK(t) of a virtual knot diagram K and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using qK(t). In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from qK(t).