Complexity : knots, colourings, and counting / D.J.A. Welsh.

Includes bibliographical references (pages 143-159) and index.

Print version record.

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.

1. The complexity of enumeration -- 1.1. Basics of complexity -- 1.2. Counting problems -- 1.3. # P-complete problems -- 1.4. Decision easy, counting hard -- 1.5. The Permanent -- 1.6. Hard enumeration problems not thought to be # P-complete -- 1.7. Self-avoiding walks -- 1.8. Toda's theorems -- 2. Knots and links -- 2.2. Tait colourings -- 2.3. Classifying knots -- 2.4. Braids and the braid group -- 2.5. The braid index and the Seifert graph of a link -- 2.6. Enzyme action -- 2.7. The number of knots and links -- 2.8. The topology of polymers -- 3. Colourings, flows and polynomials -- 3.1. The chromatic polynomial -- 3.2. The Whitney-Tutte polynomials -- 3.3. Tutte Grothendieck invariants -- 3.4. Reliability theory -- 3.5. Flows over an Abelian group -- 3.6. Ice models -- 3.7. A catalogue of invariants -- 4. Statistical physics -- 4.1. Percolation processes -- 4.2. The Ising model -- 4.3. Combinatorial interpretations -- 4.4. The Ashkin-Teller-Potts model -- 4.5. The random cluster model -- 4.6. Percolation in the random cluster model -- 5. Link polynomials and the Tait conjectures -- 5.1. The Alexander polynomial -- 5.2. The Jones polynomial and Kauffman bracket -- 5.3. The Homfly polynomial -- 5.4. The Kauffman 2-variable polynomial -- 5.5. The Tait conjectures -- 5.6. Thistlethwaite's nontriviality criterion -- 5.7. Link invariants and statistical mechanics -- 6. Complexity questions -- 6.1. Computations in knot theory -- 6.2. The complexity of the Tutte plane -- 6.3. The complexity of knot polynomials -- 6.4. The complexity of the Ising model -- 6.5. Reliability and other computations -- 7. The complexity of uniqueness and parity -- 7.1. Unique solutions -- 7.2. Unambiguous machines and one-way functions -- 7.3. The Valiant-Vazirani theorem -- 7.4. Hard counting problems not parsimonious with SAT -- 7.5. The curiosity of parity -- 7.6. Toda's theorem on parity -- 8. Approximation and randomisation -- 8.1. Metropolis methods -- 8.2. Approximating to within a ratio -- 8.3. Generating solutions at random -- 8.4. Rapidly mixing Markov chains -- 8.5. Computing the volume of a convex body -- 8.6. Approximations and the Ising model.

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