Kashaev's volume conjecture
imported>Pythagoras0님의 2020년 11월 13일 (금) 02:50 판
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
- 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
encyclopedia
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
- Alexander Kolpakov, Jun Murakami, Combinatorial decompositions, Kirillov-Reshetikhin invariants and the Volume Conjecture for hyperbolic polyhedra, http://arxiv.org/abs/1603.02380v1
- Chen, Qingtao, Kefeng Liu, and Shengmao Zhu. “Volume Conjecture for $SU(n)$-Invariants.” arXiv:1511.00658 [hep-Th, Physics:math-Ph], November 2, 2015. http://arxiv.org/abs/1511.00658.
- 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.