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 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. 
Rui Pedro Carpentier