"3-manifolds and their invariants"의 두 판 사이의 차이

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imported>Pythagoras0
 
(사용자 2명의 중간 판 19개는 보이지 않습니다)
8번째 줄: 8번째 줄:
 
* maps between aspherical 3 manifolds
 
* maps between aspherical 3 manifolds
 
* aspherical threefolds = second and higher homotopy groups vanish
 
* aspherical threefolds = second and higher homotopy groups vanish
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*  JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition<br>
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*  JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
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**  cutting M into<br>
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**  cutting M into
 
*** Seifert fibered pieces ~ non hyperbolic pieces
 
*** Seifert fibered pieces ~ non hyperbolic pieces
 
*** atoroidal pieces ~ hyperbolic pieces
 
*** atoroidal pieces ~ hyperbolic pieces
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*  Thurston's geometrization<br>
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*  Thurston's geometrization
 
** S^3, E\times S^2, Sol
 
** S^3, E\times S^2, Sol
 
** E^3, E\times H^2, SL_2
 
** E^3, E\times H^2, SL_2
 
** H^3, Nil
 
** H^3, Nil
  
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==Volume of knot complement==
 
==Volume of knot complement==
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*  Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
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* {{수학노트|url=블로흐-비그너_다이로그(Bloch-Wigner_dilogarithm)}}
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* {{수학노트|url=로바체프스키_함수}}
  
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#  KnotData[]<br> KnotData["FigureEight", "HyperbolicVolume"]<br> N[%, 20]<br>
 
  
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* Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold<br>
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* [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm] is involved<br>
 
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==a problem==
 
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*  Prove<br>$$
 
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\begin{align}
 
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\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt & =\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7})) \\
 
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& = \frac{2}{\sqrt{7}}(Cl(2\pi /7)+Cl(4\pi/7)-Cl(6\pi/7))
 
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\end{align}
 
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$$<br>
 
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* [[a log tangent integral]]<br>
 
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==invariants==
 
==invariants==
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* [[Chern-Simons gauge theory and Witten's invariant]]
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* [[Chern-Simons invariant]]
 
* Turaev-Viro invariant (related to [[6j symbols (Racha coefficient)|6j symbols]])
 
* Turaev-Viro invariant (related to [[6j symbols (Racha coefficient)|6j symbols]])
 
** Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds"
 
** Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds"
 
** Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols)
 
** Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols)
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* [[Chern-Simons invariant]]
 
 
* [[Kashaev's volume conjecture]]
 
* [[Kashaev's volume conjecture]]
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* [[Triangulations and the Bloch group]]
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* [[Ideal triangulations of 3-manifolds and the Bloch invariant]]
 
* [[Volume of hyperbolic threefolds and L-values]] and volume of knot complements
 
* [[Volume of hyperbolic threefolds and L-values]] and volume of knot complements
 
* [[Number fields and threefolds]]
 
* [[Number fields and threefolds]]
 
* [[Reidemeister torsion]]
 
* [[Reidemeister torsion]]
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==Reshetikihn, Turaev==
 
==Reshetikihn, Turaev==
  
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==software==
 
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* [http://www.geometrygames.org/SnapPea/ snappea]
 
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* [http://sourceforge.net/projects/snap-pari/ snap]
 
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* [http://www.math.utk.edu/%7Emorwen/knotscape.html http://www.math.utk.edu/~morwen/knotscape.html]
 
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==history==
 
==history==
74번째 줄: 50번째 줄:
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
  
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80번째 줄: 56번째 줄:
 
* [[Topological quantum field theory(TQFT)]]
 
* [[Topological quantum field theory(TQFT)]]
 
* [[quantum dilogarithm]]
 
* [[quantum dilogarithm]]
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* [[threefolds and their invariants]]
 
 
* [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]]
 
* [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]]
 
* [[Gieseking's constant]]
 
* [[Gieseking's constant]]
 
* [[mathematics of x^3-x+1=0]]
 
* [[mathematics of x^3-x+1=0]]
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* [[triangulations and Bloch group]]
 
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* [[volume of hyperbolic threefolds and L-values]]
 
  
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==encyclopedia==
 
==encyclopedia==
  
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* http://en.wikipedia.org/wiki/Quantum_invariant<br>
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* http://en.wikipedia.org/wiki/Quantum_invariant
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* http://ko.wikipedia.org/wiki/[http://en.wikipedia.org/wiki/Figure-eight_knot_%28mathematics%29 ]
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* http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
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* http://en.wikipedia.org/wiki/
 
  
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==books==
 
==books==
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* Saveliev, Nikolai. 1999. Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant. Walter De Gruyter Inc. http://www.amazon.com/Lectures-Topology-3-Manifolds-Introduction-Invariant/dp/3110162725
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*  Tomotada Ohtsuki [http://www.worldscibooks.com/mathematics/4746.html Quantum Invariants]
  
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* http://www.worldscibooks.com/mathematics/4746.html<br>
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* [[2010년 books and articles]]<br>
 
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==expositions==
 
==expositions==
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* Delp, Kelly, Diane Hoffoss, and Jason Fox Manning. “Problems In Groups, Geometry, and Three-Manifolds.” arXiv:1512.04620 [math], December 14, 2015. http://arxiv.org/abs/1512.04620.
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*  Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier<br>
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*  Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier
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*  Christian Blanchet, Vladimir Turaev [http://www.math.jussieu.fr/%7Eblanchet/Articles/EMP_quantum_inv.pdf Quantum Invariants of 3-manifolds]<br>
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*  Christian Blanchet, Vladimir Turaev [http://www.math.jussieu.fr/%7Eblanchet/Articles/EMP_quantum_inv.pdf Quantum Invariants of 3-manifolds]
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* Scott, Peter. 1983. “The Geometries of <math>3</math>-manifolds.” The Bulletin of the London Mathematical Society 15 (5): 401–487. doi:10.1112/blms/15.5.401.
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==articles==
 
==articles==
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* Maria, Clément, and Jonathan Spreer. “Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants.” arXiv:1512.04648 [cs, Math], December 14, 2015. http://arxiv.org/abs/1512.04648.
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* Friedl, Stefan, and Wolfgang Lück. “The L^2-Torsion Function and the Thurston Norm of 3-Manifolds.” arXiv:1510.00264 [math], October 1, 2015. http://arxiv.org/abs/1510.00264.
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* Kuperberg, Greg. “Algorithmic Homeomorphism of 3-Manifolds as a Corollary of Geometrization.” arXiv:1508.06720 [math], August 27, 2015. http://arxiv.org/abs/1508.06720.
 
* [http://arxiv.org/abs/hep-th/9811173 Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links] J.M. Borwein, D.J. Broadhurst, 1998
 
* [http://arxiv.org/abs/hep-th/9811173 Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links] J.M. Borwein, D.J. Broadhurst, 1998
 
* Gliozzi, F., and R. Tateo. 1995. Thermodynamic Bethe Ansatz and Threefold Triangulations. hep-th/9505102 (May 17). doi:doi:[http://dx.doi.org/10.1142/S0217751X96001905 10.1142/S0217751X96001905]. http://arxiv.org/abs/hep-th/9505102.
 
* Gliozzi, F., and R. Tateo. 1995. Thermodynamic Bethe Ansatz and Threefold Triangulations. hep-th/9505102 (May 17). doi:doi:[http://dx.doi.org/10.1142/S0217751X96001905 10.1142/S0217751X96001905]. http://arxiv.org/abs/hep-th/9505102.
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* Kohno, Toshitake, and Toshie Takata. "Level-Rank Duality of Witten's 3-Manifold Invariants." Progress in algebraic combinatorics 24 (1996): 243. http://tqft.net/other-papers/knot-theory/Level-rank%20duality%20-%20Kohno,%20Takata.pdf
 
* Three-manifolds and the Temperley-Lieb algebra W. B. R. Lickorish, 1991
 
* Three-manifolds and the Temperley-Lieb algebra W. B. R. Lickorish, 1991
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* [http://www.springerlink.com/content/v36272439g3g5006/ Hyperbolic manifolds and special values of Dedekind zeta-functions] Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월<br>
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* [http://www.springerlink.com/content/v36272439g3g5006/ Hyperbolic manifolds and special values of Dedekind zeta-functions] Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
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* http://dx.doi.org/10.1063/1.3085764
 
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==question and answers(Math Overflow)==
 
  
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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==blogs==
 
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*  구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
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==experts on the field==
 
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* http://arxiv.org/
 
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==links==
 
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
 
[[분류:개인노트]]
 
[[분류:개인노트]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
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[[분류:TQFT]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q526901 Q526901]
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===Spacy 패턴 목록===
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* [{'LEMMA': '3-manifold'}]

2021년 2월 17일 (수) 01:12 기준 최신판

fundamental results on three manifolds

  • Mostow-Prasad rigidity
  • geometrization


maps between threefolds

  • maps between aspherical 3 manifolds
  • aspherical threefolds = second and higher homotopy groups vanish
  • JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
    • cutting M into
      • Seifert fibered pieces ~ non hyperbolic pieces
      • atoroidal pieces ~ hyperbolic pieces
  • Thurston's geometrization
    • S^3, E\times S^2, Sol
    • E^3, E\times H^2, SL_2
    • H^3, Nil



Volume of knot complement


invariants


Reshetikihn, Turaev

history



related items


encyclopedia



books


expositions

articles

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LEMMA': '3-manifold'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=3-manifolds_and_their_invariants&oldid=51698"