"Kashaev's volume conjecture"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| − | + | ==introduction== | |
| − | + | * The hyperbolic volume of a knot complement can be calculated using the Jones polynimials of the ca | |
| + | * <math>SU(2)</math> connections on <math>S^3-K</math> should be sensitive to the flat <math>SL_2(C)</math> 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== |
| + | * <math>4_1</math> figure eight knot | ||
| + | * <math>5_2</math> | ||
| + | * <math>6_1</math> | ||
| − | |||
| − | + | ==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 <math>\langle K \rangle_N</math> | ||
| + | * 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 <math>\langle K \rangle_N</math> can be obtained from evaluating the colored Jones polynomial at the <math>N</math>-th root of unity | ||
| − | * [[quantum dilogarithm]] | + | ==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]] | ||
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| − | + | ==computational resource== | |
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxRmVVeXlVWU9xbVk/edit | ||
| − | + | ||
| + | ==encyclopedia== | ||
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
* http://en.wikipedia.org/wiki/Volume_conjecture | * http://en.wikipedia.org/wiki/Volume_conjecture | ||
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| − | < | + | ==expositions== |
| + | * Hikami, Kazuhiro. 2003. “Volume Conjecture and Asymptotic Expansion of <math>q</math>-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 | ||
| + | ** video | ||
| + | * R. M. Kashaev , [http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=5941 Faddeev's quantum dilogarithm and 3-manifold invariants], Nov 2012 | ||
| + | ** video lecture | ||
| + | * Zagier [https://docs.google.com/file/d/0B8XXo8Tve1cxbGQwMUVpQlhlREk/edit Between Number theory and topology.pdf] | ||
| + | * http://www.math.titech.ac.jp/~Jerome/090210%20workshop.pdf | ||
| + | * [http://www.math.columbia.edu/%7Edpt/speaking/hypvol.ps Hyperbolic volume and the Jones polynomial] ([http://www.math.columbia.edu/%7Edpt/speaking/hypvol.pdf PDF]), notes from a lecture at MSRI, December 2000. [http://www.math.columbia.edu/%7Edpt/speaking/Grenoble.pdf 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 | ||
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| − | + | ==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 <math>SU(n)</math>-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 | |
| − | * | + | * [http://dx.doi.org/10.1023/A:1022608131142 Proof of the volume conjecture for torus knots] |
| − | + | ** R. M. Kashaev and O. Tirkkonen, 2003 | |
| − | + | * [http://projecteuclid.org/euclid.em/1057777432 Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links] | |
| − | * | ||
| − | * [http://dx.doi.org/10.1023/A:1022608131142 Proof of the volume conjecture for torus knots] | ||
| − | ** R. M. | ||
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| − | * [http://projecteuclid.org/euclid.em/1057777432 Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links] | ||
** 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:// | + | * [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 | |
| − | * [http://arxiv.org/abs/math/0009165 On the volume conjecture for hyperbolic knots] | + | * 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. |
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| − | + | ==links== | |
| + | * [http://staff.science.uva.nl/%7Eriveen/volume_conjecture.htm Volume conjecture links and notes] | ||
| + | * [http://www.rolandvdv.nl/research.html R. van der Veen] | ||
| + | [[분류:math and physics]] | ||
| + | [[분류:TQFT]] | ||
| + | [[분류:Knot theory]] | ||
| + | [[분류:migrate]] | ||
| − | * [ | + | ==메타데이터== |
| − | * [ | + | ===위키데이터=== |
| − | + | * ID : [https://www.wikidata.org/wiki/Q7940887 Q7940887] | |
| − | + | ===Spacy 패턴 목록=== | |
| − | + | * [{'LOWER': 'volume'}, {'LEMMA': 'conjecture'}] | |
2021년 2월 17일 (수) 03:04 기준 최신판
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.
links
메타데이터
위키데이터
- ID : Q7940887
Spacy 패턴 목록
- [{'LOWER': 'volume'}, {'LEMMA': 'conjecture'}]