Complexity

Knots, Colourings and Countings

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Paperback | 11 October 1993

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172 Pages

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22.86 x 15.24 x 1.02

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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.

Industry Reviews

"...suitable for advanced graduate students and researchers in complexity theory...The list of references is long and good, and the index is useful." Computing Reviews "...an urbane presentation...of the seemingly disparate threads that will some day be the new math...the author has attained a degree of clarity and readability that few mathematicians today are capable of." Gian-Carlo Rota "...certain to be a valuable reference..." Lorenzo Traldi, Mathematical Reviews "A clear and up-to-date survey of what's known--and what's still unknown." American Mathematical Monthly

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The complexity of enumeration	p. 1
Basics of complexity	p. 1
Counting problems	p. 3
# P-complete problems	p. 6
Decision easy, counting hard	p. 10
The Permanent	p. 11
Hard enumeration problems not thought to be #P-complete	p. 14
Self-avoiding walks	p. 16
Toda's theorems	p. 17
Additional notes	p. 19
Knots and links	p. 20
Introduction	p. 20
Tait colourings	p. 23
Classifying knots	p. 25
Braids and the braid group	p. 29
The braid index and the Seifert graph of a link	p. 31
Enzyme action	p. 35
The number of knots and links	p. 37
The topology of polymers	p. 37
Additional notes	p. 40
Colourings, flows and polynomials	p. 42
The chromatic polynomial	p. 42
The Whitney-Tutte polynomials	p. 44
Tutte Grothendieck invariants	p. 46
Reliability theory	p. 48
Flows over an Abelian group	p. 49
Ice models	p. 50
A catalogue of invariants	p. 51
Additional notes	p. 53
Statistical physics	p. 54
Percolation processes	p. 54
The Ising model	p. 58
Combinatorial interpretations	p. 60
The Ashkin-Teller-Potts model	p. 62
The random cluster model	p. 64
Percolation in the random cluster model	p. 67
Additional notes	p. 70
Link polynomials and the Tait conjectures	p. 71
The Alexander polynomial	p. 71
The Jones polynomial and Kauffman bracket	p. 75
The Homfly polynomial	p. 82
The Kauffman 2-variable polynomial	p. 84
The Tait conjectures	p. 87
Thistlethwaite's nontriviality criterion	p. 90
Link invariants and statistical mechanics	p. 92
Additional notes	p. 94
Complexity questions	p. 95
Computations in knot theory	p. 95
The complexity of the Tutte plane	p. 96
The complexity of knot polynomials	p. 100
The complexity of the Ising model	p. 104
Reliability and other computations	p. 107
Additional notes	p. 109
The complexity of uniqueness and parity	p. 110
Unique solutions	p. 110
Unambiguous machines and one-way functions	p. 113
The Valiant-Vazirani theorem	p. 114
Hard counting problems not parsimonious with SAT	p. 117
The curiosity of parity	p. 118
Toda's theorem on parity	p. 122
Additional notes	p. 123
Approximation and randomisation	p. 124
Metropolis methods	p. 124
Approximating to within a ratio	p. 126
Generating solutions at random	p. 129
Rapidly mixing Markov chains	p. 130
Computing the volume of a convex body	p. 132
Approximations and the Ising model	p. 135
Some open questions	p. 137
Additional notes	p. 141
References	p. 143
Table of Contents provided by Syndetics. All Rights Reserved.

Complexity

Knots, Colourings and Countings

At a Glance

Paperback

Industry Reviews

Shipping

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