The Tutte polynomial and related polynomials

The Tutte polynomial was defined originally for graphs and extends to more generally to matroids. The many natural combinatorial interpretations of its evaluations and coefficients for graphs then translate to not obviously related combinatorial quantities in other matroids. For example, the Tutte polynomial evaluated at (2, 0) gives the number of acyclic orientations of a graph. Zaslavsky proved that the Tutte polynomial at (2, 0) also counts the number of different arrangments of sets of hyperplanes in n-dimensional Euclidean space (the underlying matroid is of a set of n-dimensional vectors over R). For another example of extending the scope of a definition, the Tutte polynomial T (G;x, y) along the hyperbola xy = 1 when G is planar specializes to the Jones polynomial of the alternating link associated with G (via the medial graph of G). Starting from the knot theory context, analogues of the Tutte polynomial have recently been defined for signed graphs (needed for encoding arbitrary links, not just alternating) and embedded graphs (in other surfaces than the plane). We shall see the connection of the Tutte polynomial and the Jones polynomial via the Kauffman bracket of a link: the deletion-contraction recurrence for the Tutte polynomial of a graph is mirrored in the skein relation for the Jones polynomial (involving local transformations of a knot). But before this we shall see with the interlace polynomial another example of this phenomenon – originally defined meaningfully only for a restricted class of graphs (namely interlace graphs, or circle graphs), its recursive definition (analogous to deletion-contraction for the Tutte polynomial) applies to any graph. Interpreting its values for graphs generally remains an open area of research. Its definition has already been extended to matroids too. This amid the world of graphs is in itself surprising, as it is the evaluation of the chromatic polynomial at −1: this is the tip of the iceberg with regard to the connection between orientations and colourings of graphs A hyperplane in n-dimensional Euclidean space is an (n−1)-dimensional flat subset (congruent to (n − 1)-dimensional space), i.e., affine subspace of dimension n − 1. Flats in Euclidean spaces are affine subspaces such as points, lines, planes,... More generally, a flat in a matroid is a subset with the property that adding any other element to it increases its rank, and a hyperplane in a matroid of rank r is a flat of rank r − 1.