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Acta Mathematica Vietnamica

, Volume 39, Issue 4, pp 649–710 | Cite as

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

  • Hitoshi MurakamiEmail author
Article

Abstract

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.

Keywords

Knot Volume conjecture Colored Jones polynomial Chern-Simons invariant Reidemeister torsion Iterated torus knot 

Mathematics Subject Classification (2010)

Primary 57M27 Secondary 57M25 57M50 58J28 

Notes

Acknowledgments

This article is prepared for the proceedings of the conference “The Quantum Topology and Hyperbolic Geometry” in Nha Trang, Vietnam, 13–17 May, 2013. I would like to thank the organizers for their hospitality.

Part of this work was done when the author was visiting the Max-Planck Institute for Mathematics, Université Paris Diderot, and the University of Amsterdam. The author thanks Christian Blanchet, Roland van der Veen, Jinseok Cho, and Satoshi Nawata for helpful discussions.

This work was supported by JSPS KAKENHI Grant Numbers 23340115, 24654041.

Supplementary material

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ESM 1 (.STY 4.62 KB)

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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