Yoshiyuki Ohyama

A Cn-MOVE FOR A KNOT AND THE COEFFICIENTS OF THE CONWAY POLYNOMIAL

It is shown that two knots can be transformed into each other by Cn-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a Cn-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a Cn-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

LOCAL MOVES ON A GRAPH IN R3

We extend a notion, an unknotting operation for knots, to a spatial embedding of a graph and study local moves on a diagram of a spatial graph.

THE GORDIAN COMPLEX 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 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,…,Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots dv(Ki, Kj) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.

DELTA LINK HOMOTOPY FOR TWO COMPONENT LINKS, II

In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ1 and δ2 classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n1 and n2 there exists a 2-component link κ such that δ1(κ) = n1 and δ2(κ) = n2 provided that at least one of n1 and n2 is even.

KNOTS WITH GIVEN FINITE TYPE INVARIANTS AND Ck-DISTANCES

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

A two dimensional lattice of knots by C 2m -moves

We consider a local move on a knot diagram, where we denote the local move by M . If two knots K1 and K2 are transformed into each other by a finite sequence of M -moves, the M -distance between K1 and K2 is the minimum number of times of M -moves needed to transform K1 into K2. A M -distance satisfies the axioms of distance. A two dimensional lattice of knots by M -moves is the two dimensional lattice graph which satisfies the following: The vertex set consists of oriented knots and for any two verticesK1 andK2, the distance on the graph from K1 to K2 coincides with the M -distance between K1 and K2, where the distance on the graph means the number of edges of the shortest path which connects the two knots. Local moves called Cn-moves are closely related to Vassiliev invariants. In this paper, we show that for any given knot K, there is a two dimensional lattice of knots by C2n-moves with the vertex K. 2010 Mathematics Subject Classification. 57M25

THE GORDIAN COMPLEX OF VIRTUAL KNOTS BY FORBIDDEN MOVES

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, …, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves dF(Ki, Kj) = 1.

KNOTS WITH GIVEN FINITE TYPE INVARIANTS AND CONWAY POLYNOMIAL

It is well-known that the coefficient of zm of the Conway polynomial is a Vassiiev invariant of order m. In this paper, we show that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K.

ON THE Cn-DISTANCE AND VASSILIEV INVARIANTS

A local move called a Cn-move is closely related to Vassiliev invariants. A Cn-distance between two knots K and L, denoted by dCn(K, L), is the minimum number of times of Cn-moves needed to transform K into L. Let p and q be natural numbers with p > q ≥ 1. In this paper, we show that for any pair of knots K1 and K2 with dCn(K1, K2) = p and for any given natural number m, there exist infinitely many knots Jj(j = 1, 2, …) such that dCn(K1, Jj) = q and dCn(Jj, K2) = p - q, and they have the same Vassiliev invariants of order less than or equal to m. In the case of n = 1 or 2, the knots Jj(j = 1, 2, …) satisfy more conditions.

Geodesic graphs for a homotopy class of virtual knots

We consider a local move, denoted by λ, on knot diagrams or virtual knot diagrams.If two (virtual) knots K1 and K2 are transformed into each other by a finite sequence of λ moves, the λ distance between K1 and K2 is the minimum number of times of λ moves needed to transform K1 into K2. By Γλ(K), we denote the set of all (virtual) knots which can be transformed into a (virtual) knot K by λ moves. A geodesic graph for Γλ(K) is the graph which satisfies the following: The vertex set consists of (virtual) knots in Γλ(K) and for any two vertices K1 and K2, the distance on the graph from K1 to K2 coincides with the λ distance between K1 and K2. When we consider virtual knots and a crossing change as a local move λ, we show that the N-dimensional lattice graph for any given natural number N and any tree are geodesic graphs for Γλ(K).