An Introduction to Knot Theory

1. Front Matter

   Pages i-x

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2. A Beginning for Knot Theory

   • W. B. Raymond Lickorish

   Pages 1-14

3. Seifert Surfaces and Knot Factorisation

   • W. B. Raymond Lickorish

   Pages 15-22

4. The Jones Polynomial

   • W. B. Raymond Lickorish

   Pages 23-31

5. Geometry of Alternating Links

   • W. B. Raymond Lickorish

   Pages 32-40

6. The Jones Polynomial of an Alternating Link

   • W. B. Raymond Lickorish

   Pages 41-48

7. The Alexander Polynomial

   • W. B. Raymond Lickorish

   Pages 49-65

8. Covering Spaces

   • W. B. Raymond Lickorish

   Pages 66-78

9. The Conway Polynomial, Signatures and Slice Knots

   • W. B. Raymond Lickorish

   Pages 79-92

10. Cyclic Branched Covers and the Goeritz Matrix

    • W. B. Raymond Lickorish

    Pages 93-102

11. The Arf Invariant and the Jones Polynomial

    • W. B. Raymond Lickorish

    Pages 103-109

12. The Fundamental Group

    • W. B. Raymond Lickorish

    Pages 110-122

13. Obtaining 3-Manifolds by Surgery on S ³

    • W. B. Raymond Lickorish

    Pages 123-132

14. 3-Manifold Invariants from the Jones Polynomial

    • W. B. Raymond Lickorish

    Pages 133-145

15. Methods for Calculating Quantum Invariants

    • W. B. Raymond Lickorish

    Pages 146-165

16. Generalisations of the Jones Polynomial

    • W. B. Raymond Lickorish

    Pages 166-178

17. Exploring the HOMFLY and Kauffman Polynomials

    • W. B. Raymond Lickorish

    Pages 179-192

18. Back Matter

    Pages 193-204

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This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space. Knots can be studied at many levels and from many points of view. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. The study of knots can be given some motivation in terms of applications in molecular biology or by reference to paral­ lels in equilibrium statistical mechanics or quantum field theory. Here, however, knot theory is considered as part of geometric topology. Motivation for such a topological study of knots is meant to come from a curiosity to know how the ge­ ometry of three-dimensional space can be explored by knotting phenomena using precise mathematics. The aim will be to find invariants that distinguish knots, to investigate geometric properties of knots and to see something of the way they interact with more adventurous three-dimensional topology. The book is based on an expanded version of notes for a course for recent graduates in mathematics given at the University of Cambridge; it is intended for others with a similar level of mathematical understanding. In particular, a knowledge of the very basic ideas of the fundamental group and of a simple homology theory is assumed; it is, after all, more important to know about those topics than about the intricacies of knot theory.

• Knot theory
• Signatur
• manifold
• quantum invariant
• topology

W.B.R. Lickorish

An Introduction to Knot Theory

"This essential introduction to vital areas of mathematics with connections to physics, while intended for graduate students, should fall within the ken of motivated upper-division undergraduates."—CHOICE

• Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and Fellow of Pembroke College,Cambridge, Cambridge, England

  W. B. Raymond Lickorish

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