Dichroic Polynomial

A Polynomial $Z_G(q,v)$ in two variables for abstract Graphs. A Graph with one Vertex has $Z=q$. Adding a Vertex not attached by any Edges multiplies the $Z$ by $q$. Picking a particular Edge of a Graph $G$, the Polynomial for $G$ is defined by adding the Polynomial of the Graph with that Edge deleted to $v$ times the Polynomial of the graph with that Edge collapsed to a point. Setting $v=-1$ gives the number of distinct Vertex colorings of the Graph. The dichroic Polynomial of a Planar Graph can be expressed as the Square Bracket Polynomial of the corresponding Alternating Link by

$$Z_G(q,v)=q^{N/2} B_{L(G)},$$

where $N$ is the number of Vertices in $G$. Dichroic Polynomials for some simple Graphs are

$$Z_{K_1} = q$$

$$Z_{K_2} = q^2+vq$$

$$Z_{K_3} = q^3+3vq^2+3v^2q+v^3.$$

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 231-235, 1994.