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\$.