Volume Conjecture for Knots 1st ed. 2018 [Mīkstie vāki]

  • Formāts: Paperback / softback, 120 pages, height x width: 235x155 mm, weight: 215 g, 18 Illustrations, color; 80 Illustrations, black and white; IX, 120 p. 98 illus., 18 illus. in color., 1 Paperback / softback
  • Sērija : SpringerBriefs in Mathematical Physics 30
  • Izdošanas datums: 27-Aug-2018
  • Izdevniecība: Springer Verlag, Singapore
  • ISBN-10: 9811311498
  • ISBN-13: 9789811311499
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  • Formāts: Paperback / softback, 120 pages, height x width: 235x155 mm, weight: 215 g, 18 Illustrations, color; 80 Illustrations, black and white; IX, 120 p. 98 illus., 18 illus. in color., 1 Paperback / softback
  • Sērija : SpringerBriefs in Mathematical Physics 30
  • Izdošanas datums: 27-Aug-2018
  • Izdevniecība: Springer Verlag, Singapore
  • ISBN-10: 9811311498
  • ISBN-13: 9789811311499
Citas grāmatas par šo tēmu:

The volume conjecture states that a certain limit of the colored Jones polynomial of a knot in the three-dimensional sphere would give the volume of the knot complement. Here the colored Jones polynomial is a generalization of the celebrated Jones polynomial and is defined by using a so-called R-matrix that is associated with the N-dimensional representation of the Lie algebra sl(2;C). The volume conjecture was first stated by R. Kashaev in terms of his own invariant defined by using the quantum dilogarithm. Later H. Murakami and J. Murakami proved that Kashaev’s invariant is nothing but the N-dimensional colored Jones polynomial evaluated at the Nth root of unity. Then the volume conjecture turns out to be a conjecture that relates an algebraic object, the colored Jones polynomial, with a geometric object, the volume.

In this book we start with the definition of the colored Jones polynomial by using braid presentations of knots. Then we state the volume conjecture and give a very elementary proof of the conjecture for the figure-eight knot following T. Ekholm. We then give a rough idea of the “proof”, that is, we show why we think the conjecture is true at least in the case of hyperbolic knots by showing how the summation formula for the colored Jones polynomial “looks like” the hyperbolicity equations of the knot complement.

We also describe a generalization of the volume conjecture that corresponds to a deformation of the complete hyperbolic structure of a knot complement. This generalization would relate the colored Jones polynomial of a knot to the volume and the Chern–Simons invariant of a certain representation of the fundamental group of the knot complement to the Lie group SL(2;C).

We finish by mentioning further generalizations of the volume conjecture.

1 Preliminaries
1(10)
1.1 Knot
1(2)
1.2 Satellite
3(5)
1.3 Braid
8(3)
2 R-Matrix, the Colored Jones Polynomial, and the Kashaev Invariant
11(16)
2.1 A Link Invariant Derived from a Yang--Baxter Operator
11(11)
2.1.1 Yang--Baxter Operator
11(3)
2.1.2 Colored Jones Polynomial
14(2)
2.1.3 Kashaev's R-Matrix
16(1)
2.1.4 Example of Calculation
16(6)
2.2 Colored Jones Polynomial via the Kauffman Bracket
22(5)
2.2.1 Kauffman Bracket
22(2)
2.2.2 Example of Calculation
24(3)
3 Volume Conjecture
27(8)
3.1 Volume Conjecture
27(1)
3.2 Figure-Eight Knot
28(2)
3.3 Torus Knot
30(5)
4 Idea of "Proof"
35(30)
4.1 Algebraic Part
35(4)
4.2 Analytic Part
39(13)
4.2.1 Integral Expression
40(2)
4.2.2 Potential Function
42(2)
4.2.3 Saddle Point Method
44(7)
4.2.4 Remaining Tasks
51(1)
4.3 Geometric Part
52(13)
4.3.1 Ideal Triangulation
52(3)
4.3.2 Cusp Triangulation
55(1)
4.3.3 Hyperbolicity Equations
56(6)
4.3.4 Complex Volumes
62(3)
5 Representations of a Knot Group, Their Chern-Simons Invariants, and Their Reidemeister Torsions
65(28)
5.1 Representations of a Knot Group
65(7)
5.1.1 Presentation
65(3)
5.1.2 Representation
68(4)
5.2 The Chern-Simons Invariant
72(10)
5.2.1 Definition
72(2)
5.2.2 How to Calculate
74(8)
5.3 Twisted SL(2; C) Reidemeister Torsion
82(11)
5.3.1 Definition
82(5)
5.3.2 How to Calculate
87(6)
6 Generalizations of the Volume Conjecture
93(20)
6.1 Complexification
93(1)
6.2 Refinement
94(6)
6.2.1 Figure-Eight Knot
94(6)
6.2.2 Torus Knot
100(1)
6.3 Parametrization
100(8)
6.3.1 Torus Knot
101(2)
6.3.2 Figure-Eight Knot
103(5)
6.4 Miscellaneous Results
108(1)
6.5 Final Remarks
109(4)
References 113(6)
Index 119