アブストラクト

We consider an infinite graph whose vertices are lattice points
in $R^2$ satisfying that two vertices are connected by an edge
if and only if Euclidean distance between the pair is equal to one.
We call it a two dimensional lattice graph.
  We consider a local move and if two knots $K_1$ and $K_2$ are transformed
into each other by a finite sequence of the local moves, we denote the
minimum number of times of the local moves needed to transform $K_1$
into $K_2$ by $d_{M}(K_{1},K_{2})$.
  A two dimensional lattice of knots by the local move is the two dimensional
lattice graph which satisfies the following:
(1) The vertex set consists of oriented knots.
(2) For any two vertices $K_1$ and $K_2$, $d(K_{1},K_{2})=d_{M}(K_{1},K_{2})$,
where $d(K_{1},K_{2})$ means the distance on the graph, that is,
the number of edges of the shortest path which connects $K_{1}$ and $K_{2}$.
  Local moves called $C_n$-moves are closely related to Vassiliev invariants.
  In this talk, we show that for any given knot $K$, there is a two dimensional
lattice of knots by $C_{2n}$-moves with the vertex $K$.