Vassiliev invariants of knots in a spatial graph

Abstract

We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R 3 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 R 3 that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

Homotopy on spatial graphs and the Sato-Levine invariant

“…Let g be a spatial embedding of G as illustrated in Figure 5.1 (1). We can see that g contains exactly one non-trivial 2-component link L = g(γ 13 ) ∪ g(γ) which is the Whitehead link, so lk(L) = 0 and a 3 (L) = 1. Thus by Theorem 5.2 we have that β ω p ,ω (g) is an edge-homotopy invariant of g and β ω p ,ω (g) = 1.…”

“…On the other hand, there are very few studies about edge (resp. vertex)-homotopy on spatial graphs [18], [9], [13], [11].…”

Realization of knots and links in a spatial graph

“…We note that the 'only if part of Theorem 2is shown in [8] and the 'if' part of Theorem 2is shown in [19]. We refer the reader to [19], [12], [13] and [11] for related results.…”

A refinement of the Conway-Gordon theorems

“…We remark here that the second coefficient of the Conway polynomial and the square of the linking number are Vassiliev invariants [21] of order 2 [2], [10]. See also [14] for general results about Vassiliev invariants of knots and links in a spatial graph.…”