An Introduction to Knot Theory

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Springer Science & Business Media, Oct 21, 1997 - Mathematics - 201 pages
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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.
  

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Contents

II
1
III
13
IV
15
V
21
VI
23
VII
30
VIII
32
IX
40
XX
103
XXI
108
XXII
110
XXIII
121
XXIV
123
XXV
132
XXVI
133
XXVII
144

X
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XI
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XII
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XIII
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XIV
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XV
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XVI
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XVII
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XVIII
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XIX
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XXVIII
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XXIX
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XXX
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XXXI
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XXXII
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XXXIII
191
XXXIV
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XXXV
199
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Page 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
Page 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.
Page 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...

References to this book

Knots
Gerhard Burde,Heiner Zieschang
Limited preview - 2003
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