Variation in Writhes of Virtual Knots under a local move

Abstract

n-writhes denoted by Jn(K) are virtual knot invariants for n ≠ 0 and are closely associated with coefficients of some polynomial invariants of virtual knots. In this work, we investigate the variations of Jn(K) under arc shift move and conclude that n-writhes Jn(K) vary randomly in the sense that it may change by any random integer value under one arc shift move. Also, for each n ≠ 0 we provide an infinite family of virtual knots which can be distinguished by n-writhes Jn(K), whereas odd writhe J(K) fails to do so.