The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

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• 作者：Hitoshi Murakami
• 关键词：Knot ; Volume conjecture ; Colored Jones polynomial ; Chern ; Simons invariant ; Reidemeister torsion ; Iterated torus knot ; Primary 57M27 ; Secondary 57M25 ; 57M50 ; 58J28
• 刊名：Acta Mathematica Vietnamica
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：39
• 期：4
• 页码：649-710
• 全文大小：1,707 KB
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• 刊物主题：Mathematics, general;
• 出版者：Springer Singapore
• ISSN：2315-4144

文摘

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.