Volume Conjecture for Knots

Name: Volume Conjecture for Knots
Brand: Springer
SKU: 9789811311499
Price: 5087.0 INR
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by Hitoshi Murakami

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ISBN13: 9789811311499
Binding: Paperback
Subject: N/A
Publisher: Springer
Publisher Imprint: Springer
Publication Date: August 27, 2018
Pages: 120
Original Price: EUR 69.99
Language: English
Edition: 2018
Item Weight: 211 grams
BISAC Subject(s): Physics / Mathematical & Computational, Topology, and Geometry / Non-Euclidean

About The 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 N^th 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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