We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, t/)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y) -plane is #P-hard except when (x-l)(y-l) = l or when (x,y) equals (

These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics held at Rutgers University. Their aim is to link together algorithmic problems arising in knot theory, statistical physics and classical combinatorics. Apart from the theory of computational complexity concerned with enumeration problems, introductions are given to several of the topics treated, such as combinatorial knot theory, randomised approximation algorithms, percolation and random cluster models. To researchers in discrete mathematics, computer science and statistical physics, this book will be of great interest, but any non-expert should find it an appealing guide to a very active area of research.

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We study various properties of the random planar graph R n , drawn uniformly at random from the class P n of all simple planar graphs on n labelled vertices. In particular, we show that the probability that R n is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability R n has linearly many vertices of a given degree, in each embedding R n has linearly many faces of a given size, and R n has exponentially many automorphisms.