Sumiko Horiuchi

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.

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.

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.

ON A BALL IN A METRIC SPACE OF KNOTS BY DELTA MOVES

We consider the metric space of all knots on which the distance is defined by delta moves. We show that for any two knots K1 and K2 with delta distance k and for any natural numbers l and m with l + m = k, the intersection of the ball of radius l centered at K1 and the ball of radius m centered at K2 contains infinitely many knots. We also consider the problem whether or not the center of a given ball is unique.

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).

Intersection of two spheres in the metric space of knots by Cn-moves

A local move called a Cn-move is closely related to Vassiliev invariants. The Cn-distance between two knots K and L, denoted by dCn(K, L), is the minimal number of Cn-moves needed to transform K into L. In the case of n ≥ 3, we show that for any pair of knots K1 and K2 with dCn(K1, K2) = 1 and for any given natural number m, there exist infinitely many knots Jj (j = 1, 2, …) such that dCn(K1, Jj) = dCn(Jj, K2) = 1 and they have the same Vassiliev invariants of order less than or equal to m.

A NUMERICAL INVARIANT FOR TWO COMPONENT SPATIAL GRAPHS

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.

ALMOST ALTERNATING KNOTS PRODUCING AN ALTERNATING KNOT

Adams et al. introduce the notion of almost alternating links; non-alternating links which have a projection whose one crossing change yields an alternating projection. For an alternating knot K, we consider the number Alm(K) of almost alternating knots which have a projection whose one crossing change yields K. We show that for any given natural number n, there is an alternating knot K with Alm(K) ≥ n.