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Volume Conjecture for Knots

Part of the SpringerBriefs in Mathematical Physics Book Series (volume 30)

DOI: 10.1007/978-981-13-1150-5

ISBN: 978-981-13-1149-9

Publisher: Springer Singapore - Springer Journals

Aug 16, 2018

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Chapter 1Preliminaries

Murakami, Hitoshi; Yokota, Yoshiyuki

Pages: 1 –10

[In this chapter we describe fundamental definitions and theorems. For details, see for example Burde et al. (Knots, extended ed., De Gruyter studies in mathematics, vol 5. De Gruyter, Berlin, 2014. MR 3156509), Lickorish (An introduction to knot theory. Graduate texts in mathematics, vol 175....
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Chapter 2R-Matrix, the Colored Jones Polynomial, and the Kashaev Invariant

Murakami, Hitoshi; Yokota, Yoshiyuki

Pages: 11 –26

[In this chapter we give definitions of the colored Jones polynomial. To do that we use a braid presentation and a knot diagram. Kashaev’s invariant is obtained as a specialization of the colored Jones polynomial.]
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Chapter 3Volume Conjecture

Murakami, Hitoshi; Yokota, Yoshiyuki

Pages: 27 –34

[In Kashaev (Lett Math Phys 39(3):269–275, 1997. MR 1434238), Kashaev proposed a conjecture that his invariant 〈K〉N defined in Kashaev (Mod Phys Lett A 10(19):1409–1418, 1995. MR 1341338) would grow exponentially with respect to N and that its growth rate would give the hyperbolic volume of the...
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Chapter 4Idea of “Proof”

Murakami, Hitoshi; Yokota, Yoshiyuki

Pages: 35 –63

[In this chapter, for a hyperbolic knot K, we explain an idea of a possible proof of the Volume Conjecture by using Kashaev’s invariant 〈K〉N of K, which is known to be the N-colored Jones polynomial JN(K, q) evaluated at q=exp2π−1N\documentclass[12pt]{minimal} \usepackage{amsmath}...
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Chapter 5Representations of a Knot Group, Their Chern–Simons Invariants, and Their Reidemeister Torsions

Murakami, Hitoshi; Yokota, Yoshiyuki

Pages: 65 –91

[In this chapter, we describe representations of the fundamental group of a knot complement to SL(2;ℂ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
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Chapter 6Generalizations of the Volume Conjecture

Murakami, Hitoshi; Yokota, Yoshiyuki

Pages: 93 –111

[In this chapter we show various generalizations of the volume conjecture. Firstly, we introduce the complexification of the conjecture by studying the imaginary part of logJN(K;exp(2π−1∕N))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...

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