THE GORDIAN COMPLEX OF KNOTS

In this paper, we define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an infinite family of distinct knots containing K such that any pair (K_i, K_j) of the member of the family, the Gordian distance d_G(K_i, K_j) = 1.