Vassiliev invariants of knots in a spatial graph

Yoshiyuki Ohyama and Kouki Taniyama

Vol. 200 (2001), No. 1, 191–205

DOI: 10.2140/pjm.2001.200.191

Abstract

We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R³ satisify certain equations that are independent of the choice of the embedding of G. By a similar observation we define certain edge-homotopy invariants and vertex-homotopy invariants of spatial graphs based on the Vassiliev invariants of the knots contained in a spatial graph. A graph G is called adaptable if, given a knot type for each cycle of G, there is an embedding of G into R³ that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

Milestones

Published: 1 September 2001