Clasp-pass moves and Kauffman polynomials of virtual knots

Abstract

Habiro showed that two knots K-1 and K-2 are related by a finite sequence of clasp-pass moves, if and only if they have the same value for Vassiliev invariants of type <3. Tsukamoto showed that, if two knots differ by a clasp-pass move then the values of the Vassiliev invariant V-K ''' (1) of degree 3 for the two knots differ by +/- 36 or 0, where V-K(t) is the Jones polynomial of a knot K. If two virtual knots are related by clasp-pass moves, then they take the same value for all Vassiliev invariants of degree <3. We extend the Tsukamoto's result to virtual knots by using a Vassiliev invariant v(3) of degree 3, which is induced from the Kauffman polynomial. We also get a lower bound for the minimal number of clasp-pass moves needed to transform K-1 to K-2, if two virtual knots K-1 and K-2 can be related by a finite sequence of clasp-pass moves