大山 淑之

Satoh and Taniguchi introduced the [Formula: see text]-writhe [Formula: see text] for each non-zero integer [Formula: see text], which is an invariant for virtual knots. The [Formula: see text]-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The values of [Formula: see text]-writhes changed by some local moves are calculated. However for the virtualization, it is unknown. In this paper, we show that for any given non-zero integer [Formula: see text] and any given integer [Formula: see text], there exists a virtual knot whose unknotting number by the virtualization is one and the value of the [Formula: see text]-writhe equals [Formula: see text]. Namely, the virtualization of a real crossing changes the value of [Formula: see text]-writhe by any given integer [Formula: see text].

We consider a local move, denoted by., on knot diagrams or virtual knot diagrams. If two (virtual) knots K-1 and K-2 are transformed into each other by a finite sequence of lambda moves, the lambda distance between K-1 and K-2 is the minimum number of times of lambda moves needed to transform K-1 into K-2. By Gamma(lambda)(K), we denote the set of all (virtual) knots which can be transformed into a (virtual) knot K by lambda moves. A geodesic graph for Gamma(lambda)(K) is the graph which satisfies the following: The vertex set consists of (virtual) knots in Gamma(lambda)(K) and for any two vertices K-1 and K-2, the distance on the graph from K-1 to K-2 coincides with the lambda distance between K-1 and K-2. When we consider virtual knots and a crossing change as a local move lambda, we show that the N-dimensional lattice graph for any given natural number N and any tree are geodesic graphs for Gamma(lambda)(K).

A local move called a C-n-move is closely related to Vassiliev invariants. The C-n-distance between two knots K and L, denoted by d(Cn) (K, L), is the minimal number of C-n-moves needed to transform K into L. In the case of n >= 3, we show that for any pair of knots K-1 and K-2 with d(Cn) (K-1, K-2) = 1 and for any given natural number m, there exist infinitely many knots J(j) (j = 1, 2,...) such that dC(n) (K1, J(j)) = d(Cn) (J(j), K-2) = 1 and they have the same Vassiliev invariants of order less than or equal to m.

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. © World Scientific Publishing Company.

We consider a local move on a knot diagram, where we denote the local move by lambda. If two knots K-1 and K-2 are transformed into each other by a finite sequence of lambda-moves, the lambda-distance between K-1 and K-2 is the minimum number of times of lambda-moves needed to transform K-1 into K-2. A. lambda-distance satisfies the axioms of distance. A two dimensional lattice of knots by lambda-moves is the two dimensional lattice graph which satisfies the following: the vertex set consists of oriented knots and for any two vertices K-1 and K-2, the distance on the graph from K-1 to K-2 coincides with the lambda-distance between K-1 and K-2, where the distance on the graph means the number of edges of the shortest path which connects the two knots. Local moves called C-n-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 C-2m-moves with the vertex K.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S-3. 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 K-0 and for any given natural number n, there exists a family of virtual knots {K-0, K-1, . . . , 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.

A local move called a C-n-move is closely related to Vassiliev invariants. A C-n-distance between two knots K and L, denoted by d(Cn) (K, L), is the minimum number of times of C-n-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 K-1 and K-2 with d(Cn) (K-1, K-2) = p and for any given natural number m, there exist infinitely many knots J(j) (j = 1, 2,...) such that d(Cn) (K-1, J(j)) = q and d(Cn) (J(j), K-2) - 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 J(j) (j = 1, 2,...) satisfy more conditions.

We define an invariant for two component spatial graphs. Although the definition of the invariant is alike a linking number, it is different from the absolute value of a linking number. We show that the invariant is not a finite type invariant.