The computational complexity of knot and matroid polynomials

Abstract

In this paper I survey the computational complexity of the Tutte plane. For example, except for a finite set of 9 points and two special curves, evaluating the Tutte polynomial is #P-hard even for planar graphs. One of these special curves is the trivial hyperbola ( x − 1)( y − 1) = 1, the other is where the Tutte polynomial specialises to the Ising partition function. Corresponding results for knot polynomials and other classes of matroids are given.