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.

Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta-equivalence. And they also gave a necessary condition for two links to be self delta-equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self C₁-equivalence and quasi self C₂-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self C_k-equivalence and quasi self C_k+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self C_k-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

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.