"Kashaev's volume conjecture"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==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 | ||
| 13번째 줄: | 15번째 줄: | ||
==history== | ==history== | ||
* 1995 Kashaev constructed knot invariants $\langle K \rangle_N$ | * 1995 Kashaev constructed knot invariants $\langle K \rangle_N$ | ||
| − | * 1997 | + | * 1997 Kashaev proposed that the asymptotic behaviour of the 1995 invariant involves the volume of the hyperbolic 3-manifold |
| − | * 2001 | + | * 2001 '''[MM01]''' Murakami-Murakami found that $\langle K \rangle_N$ can be obtained from colored Jones polynomial at the $N$-th root of unity |
| 35번째 줄: | 37번째 줄: | ||
==expositions== | ==expositions== | ||
| + | * Hikami, Kazuhiro. 2003. “Volume Conjecture and Asymptotic Expansion of $q$-Series.” Experimental Mathematics 12 (3): 319–337. http://projecteuclid.org/euclid.em/1087329235 | ||
* [http://www.youtube.com/watch?v=KszBLLJKccQ Introduction to the Volume Conjecture, Part I], by Hitoshi Murakami | * [http://www.youtube.com/watch?v=KszBLLJKccQ Introduction to the Volume Conjecture, Part I], by Hitoshi Murakami | ||
** video | ** video | ||
| 50번째 줄: | 53번째 줄: | ||
* 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. | * 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 | * 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 | ||
| − | |||
| − | |||
* [http://dx.doi.org/10.1023/A:1022608131142 Proof of the volume conjecture for torus knots] | * [http://dx.doi.org/10.1023/A:1022608131142 Proof of the volume conjecture for torus knots] | ||
** R. M. Kashaev and O. Tirkkonen, 2003 | ** R. M. Kashaev and O. Tirkkonen, 2003 | ||
| 57번째 줄: | 58번째 줄: | ||
** Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota, 2002 | ** Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota, 2002 | ||
* [http://arxiv.org/abs/math-ph/0105039 Hyperbolic Structure Arising from a Knot Invariant], 2001 | * [http://arxiv.org/abs/math-ph/0105039 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 [http://arxiv.org/abs/math/0009165 On the volume conjecture for hyperbolic knots], 2000 | * Yoshiyuki Yokota [http://arxiv.org/abs/math/0009165 On the volume conjecture for hyperbolic knots], 2000 | ||
| − | * R. M. | + | * 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. | ||
| + | |||
2013년 12월 12일 (목) 12:03 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
examples
- $4_1$
- $5_2$
- $6_1$
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 colored Jones polynomial at the $N$-th root of unity
- complex Chern-Simons theory
- quantum dilogarithm
- quantum modular forms
- Volume of hyperbolic threefolds and L-values
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
- 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.