Topological notions for Kauffman and Vogel’s polynomial

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 the skein relations: [ ] = A[ ] + B[ ] + [ ] [ ] = a[ ], [ ] = a[ ] and is defined in terms of a state-sum and the Dubrovnik polynomial for links. In [4] it is proved, in the case B = A−1 and a = A, that for a planar graph G we have [G] = 2c−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, with the variables B = A−1 and a = A, without resorting to the skein relation.