Volume Conjecture for Knots

Benefits

• Provides a short but effective introduction to quantum invariants of knots and links

• Provides a short but effective introduction to the geometry of a knot complement

• Gives the current status of the volume conjecture

Book

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 30)

1. Front Matter
Pages i-ix
2. Hitoshi Murakami, Yoshiyuki Yokota
Pages 1-10
3. Hitoshi Murakami, Yoshiyuki Yokota
Pages 11-26
4. Hitoshi Murakami, Yoshiyuki Yokota
Pages 27-34
5. Hitoshi Murakami, Yoshiyuki Yokota
Pages 35-63
6. Hitoshi Murakami, Yoshiyuki Yokota
Pages 65-91
7. Hitoshi Murakami, Yoshiyuki Yokota
Pages 93-111
8. Back Matter
Pages 113-120

Introduction

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.

Keywords

Volume conjecture Colored Jones polynomial Knot Chern-Simons invariant Hyperbolic geometry

Authors and affiliations

1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
2. 2.Department of Mathematics and Information ScienceTokyo Metropolitan UniversityTokyoJapan

Bibliographic information

• Book Title Volume Conjecture for Knots
• Authors Hitoshi Murakami
Yoshiyuki Yokota
• Series Title SpringerBriefs in Mathematical Physics
• Series Abbreviated Title SpringerBriefs in Mathematical Physics
• DOI https://doi.org/10.1007/978-981-13-1150-5
• Copyright Information The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2018
• Publisher Name Springer, Singapore
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-981-13-1149-9
• eBook ISBN 978-981-13-1150-5
• Series ISSN 2197-1757
• Series E-ISSN 2197-1765
• Edition Number 1
• Number of Pages IX, 120
• Number of Illustrations 80 b/w illustrations, 18 illustrations in colour
• Topics
• Buy this book on publisher's site
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Reviews

“This book is a very nice account of the volume conjecture for knots, a fascinating question that relates quantum invariants to hyperbolic geometry. … The book contains a lot of explicit examples and computations. I expect it will become a classical reference in the field.” (Joan Porti, zbMath 1410.57001, 2019)