Delta moves and Kauffman polynomials of virtual knots

Abstract

In 1990, Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single Delta-move. We extend the Okada's result for virtual knots by using a Vassiliev invariant v(2) of virtual knots of degree 2 which is induced from the Kauffman polynomial of a virtual knot. We show that v(2)(K-1) - v(2)(K-2) = +/- 48, if K-2 is a virtual knot obtained from a virtual knot K-1 by applying a Delta-move. From this we have a lower bound vertical bar v(2)(K-1) - v(2)(K-2)vertical bar/48 for the number of Delta-moves if two virtual knots K-1 and K-2 are related by a sequence of Delta-moves.