Skein polynomials and the tutte polynomial when x = y

This chapter surveys some graph polynomials that are based on medial graph constructions. While none of these polynomials are specializations of the Tutte polynomial, all of them coincide with the Tutte polynomial for special classes of graplis or along special curves. We give these relations to the Tlitte polynomial, as well as a number of combinatorial interpretations that derive from them.

• A brief review of vertex and graph states, and skein relations.

• Some graph and link polynomials arising from skein relations,
including: the Martin, or circuit partition, polynomial: the
Penrose polynomial; the Kauffman bracket; and transition
polynomials.

• Evaluations of the Tutte polynomial when x = y that come from
medial graph and skein polynomial connections.