  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.
