Narrow Results

We show that for any given pair of a natural number n and a knot K, there are infinitely many knots J_m (m=1,2,…) such that their Vassiliev invariants of order less than or equal to n coincide with those of K and that each J_m has C_k-distance 1 (k≠q 2, k=1, …, n) and C₂-distance 2 from the knot K. The C_k-distance means the minimum number of C_k-moves which transform one knot into the other.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³ by using "a crossing change". In this paper, we define the C_k-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K₀ and any given natural number n, there exists a family of knots {K₀, K₁,…, K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the C_k-distance d_Ck(K_i, K_j) = 1.