Topological notions for Kauffman and Vogel's polynomial

Details Topological notions for Kauffman and Vogel's polynomial Topological notions for Kauffman and Vogel's polynomial

2002-04-16

arxiv.org/abs/math/0204207v1

Mathematics

Geometric Topology

12 pages, 5 figures and many eps files for the skein relations. To appear in J. Knot Theory Ramifications

Scientific paper

In [2] 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, satisfying three skein relations, and is defined in terms of a state-sum and the Dubrovnik polynomial for links. In previous work by the author it is proved, in the case B=A^{-1} and a=A, 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. In this paper we will show how we can calculate the polynomial for embedded graphs, with the variables B=A^{-1} and a=A, without resorting to the skein relation.

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