"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<br>
+
* 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<br>
+
* <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==
 
==history==
−
* 1995 Kashaev
+
* 1995 Kashaev constructed knot invariants <math>\langle K \rangle_N</math>
−
* 1997 ?
+
* 1997 Kashaev proposed that the asymptotic behaviour of the 1995 invariant involves the volume of the hyperbolic 3-manifold
−
* 2001(?) Murakami
+
* 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
−
 
 
  
 
==related items==
 
==related items==
−
 
+
* [[A-polynomial]]
 
* [[quantum dilogarithm]]
 
* [[quantum dilogarithm]]
 +
* [[Chern-Simons invariant]]
 +
* [[complex Chern-Simons theory]]
 
* [[quantum modular forms]]
 
* [[quantum modular forms]]
 
* [[Volume of hyperbolic threefolds and L-values]]
 
* [[Volume of hyperbolic threefolds and L-values]]
 +
* [[Holography and volume conjecture]]
  
  
29번째 줄: 57번째 줄:
  
 
==expositions==
 
==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.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.<br>
+
* [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. <br>
+
*  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, 2008, An introduction to the volume conjecture and its generalizations
 
* H. Murakami, A quantum introduction to knot theory
 
* H. Murakami, A quantum introduction to knot theory
38번째 줄: 71번째 줄:
  
 
==articles==
 
==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
 
* 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://projecteuclid.org/euclid.em/1087329235 Volume Conjecture and Asymptotic Expansion of q-Series]<br>
+
* [http://dx.doi.org/10.1023/A:1022608131142 Proof of the volume conjecture for torus knots]
−
** Kazuhiro Hikami, Experiment. Math. Volume 12, Number 3 (2003), 319-338
 
−
* [http://dx.doi.org/10.1023/A:1022608131142 Proof of the volume conjecture for torus knots]<br>
 
 
** R. M. Kashaev and O. Tirkkonen, 2003
 
** 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://projecteuclid.org/euclid.em/1057777432 Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links]<br>
 
 
** 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
−
* [http://dx.doi.org/10.1007/BF02392716 The colored Jones polynomials and the simplicial volume of a knot]<br>
+
* '''[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.
−
** J.Murakami, H.Murakami,, Acta Math. 186 (2001), 85–104
+
* Yoshiyuki Yokota [http://arxiv.org/abs/math/0009165 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.
  
−
* [http://arxiv.org/abs/math/0009165 On the volume conjecture for hyperbolic knots]<br>
+
==links==
−
** Yoshiyuki Yokota, 2000
+
* [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]]
  
−
* [http://dx.doi.org/10.1023/A:1007364912784 The hyperbolic volume of knots from quantum dilogarithm]<br>
+
==메타데이터==
−
** R. M. Kashaev, 1996
+
===위키데이터===
−
 
+
* ID :  [https://www.wikidata.org/wiki/Q7940887 Q7940887]
−
[[분류:math and physics]]
+
===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

related items


computational resource


encyclopedia


expositions


articles

links

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'volume'}, {'LEMMA': 'conjecture'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Kashaev%27s_volume_conjecture&oldid=51900"