From planar graphs to embedded graphs -- a new approach to Kauffman and Vogel's polynomial (Q2706856)

The author considers Kauffman and Vogel's polynomial for four-valent graphs embedded in the three-dimensional Euclidean space. He makes use of the graphical calculus [\textit{L. H. Kauffman} and \textit{P. Vogel}, ibid. 1, No. 1, 59-104 (1992; Zbl 0795.57001)] to show that the above polynomial of a planar graph can be calculated recursively from that of planar graphs with less vertices. Then he gives a direct proof of the uniqueness of the (normalized) polynomial restricted to planar graphs. Moreover, the polynomial for a four-valent rigid vertex embedded graph can be expressed as a weighted sum of polynomials of planar graphs. In a special case it is proved that the polynomial for a planar graph depends only on the number of its connected components and on the number of its vertices. As a consequence, a necessary condition (but not sufficient in general) for an embedded graph to be ambient isotopic to a planar graph is given. In the appendix, it is shown that the graphical calculus implies the invariance of the polynomial up to regular isotopies.