2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63835In [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.12 pages, 5 figures and many eps files for the skein relations. To appear in J. Knot Theory RamificationsGeometric Topology57M15; 05C10; 57M27text