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.
ASJC Scopus subject areas
- 数学 (全般)