Narrow Results

A lower bound of the Gordian distance is presented in terms of the Blanchfield pairing. Our approach, in particular, allows us to show at least for $195$ pairs of unoriented nontrivial prime knots with up to $10$ crossings that their Gordian distance is equal to $3$, most of which are difficult to treat otherwise.

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.

Some evaluations of the Gordian distance from a knot to another knot are given by using three polynomial invariants called the HOMFLY polynomial, the Jones polynomial and the Q-polynomial. Furthermore, Gordian distances between a lot of pairs of knots are determined.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³. 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 K₀ and for any given natural number n, there exists a family of virtual knots {K₀, K₁,…,K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the Gordian distance of virtual knots d_v(K_i, K_j) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³. In this paper, we define the Gordian complex of virtual knots by using forbidden moves. We show that for any virtual knot K₀ and for any given natural number n, there exists a family of virtual knots {K₀, K₁, …, K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves d_F(K_i, K_j) = 1.

The unknotting number of a welded knot is considered. First, we obtain an upper-bound of the unknotting number of a welded knot by using the warping degree method. Further, we introduce a standard welded torus knot with welded datum and obtain an upper bound of the unknotting number by an algorithm with warping degree method. Secondly, we get a lower bound of the unknotting number of a welded knot by Alexander quandle colorings. Finally, we give a definition of Gordian distance for welded knots analogous to the classical case.

In this paper, the $\overline{♯}$-move is defined. We show that for any knot $K_0$, there exists an infinite family of knots $\{K_0,K_1,\dots \}$ such that the Gordian distance $d(K_i,K_j)=1$ and pass-move-Gordian distance $d_p(K_i,K_j)=1$ for any $i\ne j$. We also show that there is another infinite family of knots $\{K_0^{\prime },K_1^{\prime },\dots \}$ (where $K_0^{\prime }=K_0$) such that the $\overline{♯}$-move-Gordian distance $d_{\overline{♯}}(K_i^{\prime },K_j^{\prime })=1$ and $H(n)$-Gordian distance $d_{H(n)}(K_i^{\prime },K_j^{\prime })=1$ for any $i\ne j$ and all $n\ge 2$.

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in ${𝕊}^3$. Local moves play an important role in defining knot invariants. There are many local moves known as unknotting operations for knots. In this paper, we discuss the 4-move operation. We show that for any knot $K_0$ and for any given natural number $n$, there exists a family of knots $\{K_0,K_1,\dots ,K_n\}$ such that for any pair $(K_i,K_j)$ of distinct elements of the family, the Gordian distance of knots by 4-move is $d_f(K_i,K_j)=1$. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes.