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Volume Conjecture for Knots

  • Book
  • © 2018

Overview

Authors:
  1. Hitoshi Murakami

    1. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

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  2. Yoshiyuki Yokota

    1. Department of Mathematics and Information Science, Tokyo Metropolitan University, Tokyo, Japan

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    You can also search for this author in PubMed   Google Scholar

  • 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)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Preliminaries

    • Hitoshi Murakami, Yoshiyuki Yokota
    Pages 1-10

  3. R-Matrix, the Colored Jones Polynomial, and the Kashaev Invariant

    • Hitoshi Murakami, Yoshiyuki Yokota
    Pages 11-26

  4. Volume Conjecture

    • Hitoshi Murakami, Yoshiyuki Yokota
    Pages 27-34

  5. Idea of “Proof”

    • Hitoshi Murakami, Yoshiyuki Yokota
    Pages 35-63

  6. Representations of a Knot Group, Their Chern–Simons Invariants, and Their Reidemeister Torsions

    • Hitoshi Murakami, Yoshiyuki Yokota
    Pages 65-91

  7. Generalizations of the Volume Conjecture

    • Hitoshi Murakami, Yoshiyuki Yokota
    Pages 93-111

  8. Back Matter

    Pages 113-120
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Keywords

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.

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)

Authors and Affiliations

  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Hitoshi Murakami

  • 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

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