Authors:
- 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
Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 30)
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Table of contents (6 chapters)
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Front Matter
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Back Matter
About this book
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.
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Authors and Affiliations
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Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Hitoshi Murakami
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Department of Mathematics and Information Science, Tokyo Metropolitan University, Tokyo, Japan
Yoshiyuki Yokota
Bibliographic Information
Book Title: Volume Conjecture for Knots
Authors: Hitoshi Murakami, Yoshiyuki Yokota
Series Title: SpringerBriefs in Mathematical Physics
DOI: https://doi.org/10.1007/978-981-13-1150-5
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2018
Softcover ISBN: 978-981-13-1149-9Published: 27 August 2018
eBook ISBN: 978-981-13-1150-5Published: 15 August 2018
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: Mathematical Physics, Topology, Hyperbolic Geometry