Volume conjecture for knots / Hitoshi Murakami, Yoshiyuki Yokota.
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-mat...
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Full Text (via Springer) |
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| Main Authors: | , |
| Format: | eBook |
| Language: | English |
| Published: |
Singapore :
Springer,
2018.
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| Series: | SpringerBriefs in mathematical physics ;
v. 30. |
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Table of Contents:
- Intro; Preface; Contents; Acronyms; 1 Preliminaries; 1.1 Knot; 1.2 Satellite; 1.3 Braid; 2 R-Matrix, the Colored Jones Polynomial, and the Kashaev Invariant; 2.1 A Link Invariant Derived from a Yang-Baxter Operator; 2.1.1 Yang-Baxter Operator; 2.1.2 Colored Jones Polynomial; 2.1.3 Kashaev's R-Matrix; 2.1.4 Example of Calculation; 2.2 Colored Jones Polynomial via the Kauffman Bracket; 2.2.1 Kauffman Bracket; 2.2.2 Example of Calculation; 3 Volume Conjecture; 3.1 Volume Conjecture; 3.2 Figure-Eight Knot; 3.3 Torus Knot; 4 Idea of ̀̀Proof''; 4.1 Algebraic Part; 4.2 Analytic Part.
- 4.2.1 Integral Expression4.2.2 Potential Function; 4.2.3 Saddle Point Method; 4.2.4 Remaining Tasks; 4.3 Geometric Part; 4.3.1 Ideal Triangulation; 4.3.2 Cusp Triangulation; 4.3.3 Hyperbolicity Equations; 4.3.4 Complex Volumes; 5 Representations of a Knot Group, Their Chern-Simons Invariants, and Their Reidemeister Torsions; 5.1 Representations of a Knot Group; 5.1.1 Presentation; 5.1.2 Representation; 5.2 The Chern-Simons Invariant; 5.2.1 Definition; 5.2.2 How to Calculate; 5.3 Twisted SL(2; C) Reidemeister Torsion; 5.3.1 Definition; 5.3.2 How to Calculate.
- 6 Generalizations of the Volume Conjecture6.1 Complexification; 6.2 Refinement; 6.2.1 Figure-Eight Knot; 6.2.2 Torus Knot; 6.3 Parametrization; 6.3.1 Torus Knot; 6.3.2 Figure-Eight Knot; 6.4 Miscellaneous Results; 6.5 Final Remarks; References; Index.