In [4] 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:and is defined in terms of a state-sum and the Dubrovnik polynomial for links. Using the graphical calculus of [4] it is shown that the polynomial of a planar graph can be calculated recursively from that of planar graphs with less vertices, which also allows the polynomial of an embedded graph to be calculated without resorting to links. The same approach is used to give a direct proof of uniqueness of the (normalized) polynomial restricted to planar graphs. In the case B = A −1 and a = A, it is proved 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 1 From planar graphs to embedded graphs 2 G and v is the number of vertices of G. As a corollary, a necessary, but not sufficient, condition is obtained for an embedded graph to be ambient isotopic to a planar graph. In an appendix it is shown that, given a polynomial for planar graphs satisfying the graphical calculus, and imposing the first skein relation above, the polynomial extends to a rigid vertex regular isotopy invariant for embedded graphs, satisfying the remaining skein relations. Thus, when existence of the planar polynomial is guaranteed, this provides a direct way, not depending on results for the Dubrovnik polynomial, to show consistency of the polynomial for embedded graphs.

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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: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] = 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, with the variables B = A −1 and a = A, without resorting to the skein relation.

We prove that any triangulation of a 2-dimensional sphere with a proper strict 4-coloring on its vertices can be seen as the boundary of a triangulation of a 3-dimensional disk with the same vertices in such a way that the 4-coloring remains proper.

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