Narrow Results

By the works of Gusarov [2] and Habiro [3], it is known that a local move called the C_n move is strongly related to Vassiliev invariants of order less than n. The coefficient of the zⁿ term in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and K^C_n the set of knots obtained from a knot K by a single C_n move. Let be the set of the Conway polynomials for a set of knots . Our main result is the following: There exists a pair of knots K₁, K₂ such that ∇K₁ = ∇K₂ and . In other words, the C_n Gordian complex is not homogeneous with respect to Conway polynomial.

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.

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 $n\ge 2$.

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in ${𝕊}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

In this paper, we study the Gordian metric on the set of all theta-curves and give a lower bound of it. We define the Gordian complex of theta-curves, which is a simplicial complex whose vertices consist of all theta-curves in the 3-dimensional Euclidean space ${ℝ}^3$. We show that for any given theta-curve $\mathrm{\Theta }$, there exists an infinite family of theta-curves containing $\mathrm{\Theta }$ such that the Gordian distance between any two distinct members of this family is equal to one.

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.