Dichroic Polynomial

A polynomial  in two variables for abstract graphs. A graph with one graph vertex has . Adding a graph vertex not attached by any graph edges multiplies the  by . Picking a particular graph edge of a graph , the polynomial for  is defined by adding the polynomial of the graph with that graph edge deleted to  times the polynomial of the graph with that graph edge collapsed to a point.

Setting  gives the chromatic number of the graph. The dichroic polynomial of a planar graph can be expressed as the square bracket polynomial of the corresponding alternating link by

(1)

where  is the number of graph vertices in . Dichroic polynomials for some simple graphs are

			(2)
			(3)
			(4)

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.