Kashaev's volume conjecture

introduction

• The hyperbolic volume of a knot complement can be calculated using the Jones polynimials of the ca
• $SU(2)$ connections on $S^3-K$ should be sensitive to the flat $SL_2(C)$ connection defining its hyperbolic structure

examples

• $4_1$
• $5_2$
• $6_1$

history

• 1995 Kashaev constructed knot invariants $\langle K \rangle_N$
• 1997 ?
• 2001(?) Murakami-Murakami found that $\langle K \rangle_N$ can be obtained from colored Jones polynomial

related items

• complex Chern-Simons theory
• quantum dilogarithm
• quantum modular forms
• Volume of hyperbolic threefolds and L-values

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxRmVVeXlVWU9xbVk/edit

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Volume_conjecture

expositions

• R. M. Kashaev , Faddeev's quantum dilogarithm and 3-manifold invariants, Nov 2012
  • video lecture
• Zagier Between Number theory and topology.pdf
• http://www.math.titech.ac.jp/~Jerome/090210%20workshop.pdf
• Hyperbolic volume and the Jones polynomial (PDF), notes from a lecture at MSRI, December 2000. Earlier notes (covering more material) from a lecture series at the Grenoble summer school “Invariants des noeuds et de variétés de dimension 3”, June 1999.
• Murakami, Hitoshi. 2010. An Introduction to the Volume Conjecture. 1002.0126 (January 31). http://arxiv.org/abs/1002.0126.
• H. Murakami, 2008, An introduction to the volume conjecture and its generalizations
• H. Murakami, A quantum introduction to knot theory

articles

• Dimofte, Tudor Dan. 2010. “Refined BPS Invariants, Chern-Simons Theory, and the Quantum Dilogarithm”. Phd, California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05142010-131147918.
• Generalized volume conjecture and the A-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function , 2007 http://dx.doi.org/10.1016/j.geomphys.2007.03.008
• Volume Conjecture and Asymptotic Expansion of q-Series
  • Kazuhiro Hikami, Experiment. Math. Volume 12, Number 3 (2003), 319-338
• Proof of the volume conjecture for torus knots
  • R. M. Kashaev and O. Tirkkonen, 2003
• Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links
  • Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota, 2002
• Hyperbolic Structure Arising from a Knot Invariant, 2001
• J.Murakami, H.Murakami, The colored Jones polynomials and the simplicial volume of a knot Acta Math. 186 (2001), 85–104
• Yoshiyuki Yokota On the volume conjecture for hyperbolic knots, 2000
• R. M. Kashaev The hyperbolic volume of knots from quantum dilogarithm, 1996

links

• Volume conjecture links and notes