Topological notions for Kauffman and Vogel's polynomial

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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: [Formula: see text] 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 garph 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.