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
• hyperbolic volume is closely related to the Cherm-Simons invariant
• volume conjecture has its complexified version

Kashaev invariant

• invariant of a link using the R-matrix
• calculate the limit of the Kashaev invariant
• related with the colored Jones polynomial

optimistic limit

• volume conjecture
• idea of the optimistic limit

examples

• $4_1$ figure eight knot
• $5_2$
• $6_1$

known examples

• figure eight knot
• Borromean ring
• torus knots
• whitehead chains
• all links of zero volume
• twist knows is (almost) done

history

• 1995 Kashaev constructed knot invariants $\langle K \rangle_N$
• 1997 Kashaev proposed that the asymptotic behaviour of the 1995 invariant involves the volume of the hyperbolic 3-manifold
• 2001 [MM01] Murakami-Murakami found that $\langle K \rangle_N$ can be obtained from evaluating the colored Jones polynomial at the $N$-th root of unity

related items

• A-polynomial
• quantum dilogarithm
• Chern-Simons invariant
• complex Chern-Simons theory
• quantum modular forms
• Volume of hyperbolic threefolds and L-values
• Holography and volume conjecture

computational resource

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

encyclopedia

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

expositions

• Hikami, Kazuhiro. 2003. “Volume Conjecture and Asymptotic Expansion of $q$-Series.” Experimental Mathematics 12 (3): 319–337. http://projecteuclid.org/euclid.em/1087329235
• Introduction to the Volume Conjecture, Part I, by Hitoshi Murakami
  • video
• 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

• Fernandez-Lopez, Manuel, and Eduardo Garcia-Rio. “On Gradient Ricci Solitons with Constant Scalar Curvature.” arXiv:1409.3359 [math], September 11, 2014. http://arxiv.org/abs/1409.3359.
• Murakami, Jun. 2014. “From Colored Jones Invariants to Logarithmic Invariants.” arXiv:1406.1287 [math], June. http://arxiv.org/abs/1406.1287.
• Gang, Dongmin, Nakwoo Kim, and Sangmin Lee. “Holography of Wrapped M5-Branes and Chern-Simons Theory.” arXiv:1401.3595 [hep-Th], January 15, 2014. http://arxiv.org/abs/1401.3595.
• 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
• 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
• [MM01] Murakami, Hitoshi, and Jun Murakami. 2001. “The Colored Jones Polynomials and the Simplicial Volume of a Knot.” Acta Mathematica 186 (1): 85–104. doi:10.1007/BF02392716.
• Yoshiyuki Yokota On the volume conjecture for hyperbolic knots, 2000
• Kashaev, R. M. 1997. “The Hyperbolic Volume of Knots from the Quantum Dilogarithm.” Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics 39 (3): 269–275. doi:10.1023/A:1007364912784.
• Kashaev, R. M. 1995. “A Link Invariant from Quantum Dilogarithm.” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics 10 (19): 1409–1418. doi:10.1142/S0217732395001526.

links

• Volume conjecture links and notes
• R. van der Veen