Twist moves and the affine index polynomials of virtual knots

In this paper, we give a necessary condition for two virtual knots to be related by a finite sequence of twist moves by using the affine index polynomial, which is a Vassiliev invariant of degree 1. Trapp showed that a numerical Vassiliev invariant of degree n has a polynomial growth of degree ≤ n on a twist sequence of knots, which can be extended to a twist sequence of virtual knots. We calculate the growth of the affine index polynomial for a twist sequence of virtual knots and find the difference of the affine index polynomials of two virtual knots, which are related by a twist move. Moreover, we give a lower bound for the number of twist moves needed to transform K to K^' if K and K^' are virtual knots related by a finite sequence of twist moves.