THE GORDIAN COMPLEX OF VIRTUAL KNOTS

摘要

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S~3. In this paper, we define the Gordian complex of virtual knots which is a simplicial complex whose vertices consist of all virtual knots by using the local move which makes a real crossing a virtual crossing. We show that for any virtual knot K_0 and for any given natural number n, there exists a family of virtual knots {K_0, K_1,..., K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the Gordian distance of virtual knots dv(K_i, K_j) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.