Complexity : knots, colourings and counting /

Complexity : knots, colourings and counting /

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Bibliographic Details
Author / Creator:Welsh, D. J. A.
Imprint:Cambridge ; New York : Cambridge University Press, 1993.
Description:viii, 163 p. : ill. ; 23 cm.
Language:English
Series:London Mathematical Society lecture note series 186
Subject:
Computational complexity
Knot theory
Link theory
Combinatorial analysis.
Computational complexity.
Knot theory.
Statistical physics.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/1500054
Hidden Bibliographic Details
Other authors / contributors:Advanced Research Institute of Discrete Applied Mathematics (6th : 1991 : Rutgers University)
ISBN:0521457408
Notes:"based on a series of lectures ... at the Advanced Research Institute of Discrete Applied Mathematics (ARIDAM VI) in June 1991"--p. vii.
Includes bibliographical references (p. 143-159) and index.
Table of Contents:
  • 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
  • 1.9. Additional notes
  • 2. Knots and links
  • 2.1. Introduction
  • 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
  • 2.9. Additional notes
  • 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
  • 3.8. Additional notes
  • 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
  • 4.7. Additional notes
  • 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
  • 5.8. Additional notes
  • 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
  • 6.6. Additional notes
  • 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
  • 7.7. Additional notes
  • 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
  • 8.7. Some open questions
  • 8.8. Additional notes
  • References

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