An Introduction to Knot Theory

W.B.Raymond Lickorish

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English

Springer-Verlag New York Inc.
12 October 2012

Groups & group theory; Topology; Mathematical physics

Series: Graduate Texts in Mathematics

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.

By: W.B.Raymond Lickorish
Imprint: Springer-Verlag New York Inc.
Country of Publication: United States
Edition: Softcover reprint of the original 1st ed. 1997
Volume: 175
Dimensions: Height: 235mm, Width: 155mm, Spine: 11mm
Weight: 343g
ISBN: 9781461268697
ISBN 10: 1461268699
Series: Graduate Texts in Mathematics
Pages: 204
Publication Date: 12 October 2012
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

1. A Beginning for Knot Theory.- Exercises.- 2. Seifert Surfaces and Knot Factorisation.- Exercises.- 3. The Jones Polynomial.- Exercises.- 4. Geometry of Alternating Links.- Exercises.- 5. The Jones Polynomial of an Alternating Link.- Exercises.- 6. The Alexander Polynomial.- Exercises.- 7. Covering Spaces.- Exercises.- 8. The Conway Polynomial, Signatures and Slice Knots.- Exercises.- 9. Cyclic Branched Covers and the Goeritz Matrix.- Exercises.- 10. The Arf Invariant and the Jones Polynomia.- Exercises.- 11. The Fundamental Group.- Exercises.- 12. Obtaining 3-Manifolds by Surgery on S3.- Exercises.- 13. 3-Manifold Invariants From The Jones Polynomial.- Exercises.- 14. Methods for Calculating Quantum Invariants.- Exercises.- 15. Generalisations of the Jones Polynomial.- Exercises.- 16. Exploring the HOMFLY and Kauffman Polynomials.- Exercises.- References.

Reviews for An Introduction to Knot Theory

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