About the Kauffman and Vogel's polynomial
by
Rui Pedro Carpentier
Instituto Superior Técnico

In 1990 Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a, which coincides, under the change of variables z=A-B, with the two-variable Kauffman polynomial when restricted to links. In this talk it would aborded a simplified version of this polynomial (fixing B=A^-1 and a=A) and show that for a planar graph G we have [G]=2^c-1(-A-A^-1)^v , where c is the number of connected components of G and v is the number of vertices of G. Thus it gives a good test of the planarity for spatial four-valent graphs.

Date received: July 7, 2006