A polynomial invariant of trivalent graphs that is related to the Jones polynomial of knots

Tutte discovered a polynomial derived from graphs that gives valuable information about the graph. In this talk, I will describe a simple-to-compute polynomial invariant of a trivalent graph with a perfect matching (think: the formula for computing the Tutte polynomial or the Kauffman bracket of a link). This polynomial invariant, called the 2-factor polynomial, counts the number of 2-factors of the graph that contain the perfect matching edges. We will calculate some examples and show some implications of these counts. In particular, we will explain how this polynomial is related to the Jones polynomial and how it can be generalized to compute all of the 3-edge colorings of a trivalent graph.