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

Abstract

In \cite{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 $a$, satisfies the skein relation: $$ [\psdiag{2}{6}{overcross}]=A [\psdiag{2}{6}{horlines}]+B [\psdiag{2}{6}{verlines}]+[\psdiag{2}{6}{vertex}]$$ and is defined in terms of a state-sum and the Dubrovnik polynomial for links. Using the graphical calculus of \cite{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 $G$ 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.