Volume conjecture for knots / Hitoshi Murakami, Yoshiyuki Yokota.
Murakami, Hitoshi, author.; Yokota, Yoshiyuki, author.
2018
QA612.2
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Title
Volume conjecture for knots / Hitoshi Murakami, Yoshiyuki Yokota.
Author
Murakami, Hitoshi, author.
ISBN
9789811311505 (electronic book)
9811311501 (electronic book)
9789811311499
9811311498
Published
Singapore : Springer, 2018.
Language
English
Description
1 online resource.
Call Number
QA612.2
Dewey Decimal Classification
514/.2242
Summary
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.-- Provided by publisher.
Bibliography, etc. Note
Includes bibliographical references and index.
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Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed August 20, 2018).
Added Author
Yokota, Yoshiyuki, author.
Series
SpringerBriefs in mathematical physics ; v. 30.
Available in Other Form
Print version: 9789811311499
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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

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