An Introduction to Knot Theory

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Springer Science & Business Media, 1997 M10 3 - 201 páginas
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.
 

Contenido

A Beginning for Knot Theory
1
Seifert Surfaces and Knot Factorisation
15
Geometry of Alternating Links
32
The Jones Polynomial of an Alternating Link
41
The Alexander Polynomial
49
Covering Spaces
66
Cyclic Branched Covers and the Goeritz Matrix
93
The Fundamental Group
110
Obtaining 3Manifolds by Surgery on S3
123
Methods for Calculating Quantum Invariants
146
Generalisations of the Jones Polynomial
166
Exploring the HOMFLY and Kauffman Polynomials
179
References
193
93
196
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Página 199 - IX (1987) 5-9. [134] W. Whitten. Knot complements and groups, Topology 26 (1987) 41-44. [135] E. Witten. Quantum field theory and Jones
Página 199 - S. Yamada. A topological invariant of spatial regular graphs, Knots 90, Ed. A. Kawauchi, de Gruyter, (1992) 447-454. [138] Y. Yokota. On quantum SU(2) invariants and generalised bridge numbers of knots, Math.
Página 199 - Y. Yokota. Skeins and quantum SU(N) invariants of 3-manifolds, Math. Ann. 307 (1997) 109-138. [140] EC Zeeman. Unknotting combinatorial balls, Ann. of Math. 78 (1963) 501-526. Index 6j-symbols, 154 (n + l)-ballfi"+l /(") e TLn, 136 n -dimensional sphere S...

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