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

수학노트
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(사용자 3명의 중간 판 56개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">introduction</h5>
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==fundamental results on three manifolds==
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* Mostow-Prasad rigidity
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* geometrization
  
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* volume of knot complements
 
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* Chern-Simons invariant of manifolds
 
  
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==maps between threefolds==
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* maps between aspherical 3 manifolds
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* aspherical threefolds = second and higher homotopy groups vanish
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*  JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
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**  cutting M into
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*** Seifert fibered pieces ~ non hyperbolic pieces
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*** atoroidal pieces ~ hyperbolic pieces
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*  Thurston's geometrization
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** S^3, E\times S^2, Sol
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** E^3, E\times H^2, SL_2
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** H^3, Nil
  
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<h5 style="line-height: 2em; margin: 0px;">Volume of knot complement</h5>
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# KnotData[]<br> KnotData["FigureEight", "HyperbolicVolume"]<br> N[%, 20]<br>
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==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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* [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm]<br>
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==invariants==
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* [[Chern-Simons gauge theory and Witten's invariant]]
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* [[Chern-Simons invariant]]
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* Turaev-Viro invariant (related to [[6j symbols (Racha coefficient)|6j symbols]])
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** Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds"
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** Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols)
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* [[Kashaev's volume conjecture]]
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* [[Ideal triangulations of 3-manifolds and the Bloch invariant]]
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* [[Volume of hyperbolic threefolds and L-values]] and volume of knot complements
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* [[Number fields and threefolds]]
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* [[Reidemeister torsion]]
  
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복소이차수체의 [http://pythagoras0.springnote.com/pages/4533335 데데킨트 제타함수]
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==Reshetikihn, Turaev==
  
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<math>\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})</math>
 
  
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<math>\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})</math>
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<math>\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))</math>
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==history==
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#  L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]<br> N[Sum[JacobiSymbol[a, 7]*L[Exp[2 I*Pi*a/7]], {a, 1, 6}], 20]<br> N[L[Exp[2 I*Pi/7]] + L[Exp[4 I*Pi/7]] - L[Exp[6 I*Pi/7]], 20]<br>
 
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<h5 style="line-height: 2em; margin: 0px;">an open problem</h5>
 
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*  Prove<br><math>\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}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))</math><br>
 
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* [[5478337|an open problem in integration]]<br>
 
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<h5>software</h5>
 
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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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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">history</h5>
 
  
 
* 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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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">related items</h5>
 
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* [[4667393|dilogarithm and Nahm's conjecture (mathematica)]]<br>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
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* http://ko.wikipedia.org/wiki/
 
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* [http://en.wikipedia.org/wiki/Figure-eight_knot_%28mathematics%29 http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)]
 
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* http://en.wikipedia.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">books</h5>
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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[[4909919|]]
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
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* [http://arxiv.org/abs/hep-th/9811173 Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links]<br>
 
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** J.M. Borwein, D.J. Broadhurst, 1998
 
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* [[2010년 books and articles|논문정리]]
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html][http://www.ams.org/mathscinet ]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* http://dx.doi.org/10.1063/1.3085764
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">question and answers(Math Overflow)</h5>
 
  
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* http://mathoverflow.net/search?q=
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==related items==
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* http://mathoverflow.net/search?q=
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* [[Topological quantum field theory(TQFT)]]
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* [[quantum dilogarithm]]
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* [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]]
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* [[Gieseking's constant]]
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* [[mathematics of x^3-x+1=0]]
  
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==encyclopedia==
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs</h5>
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* http://en.wikipedia.org/wiki/Quantum_invariant
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* http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
  
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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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==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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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">experts on the field</h5>
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* http://arxiv.org/
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==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
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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==
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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.
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* [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
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* 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
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* 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
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">links</h5>
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[[분류:개인노트]]
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[[분류:math and physics]]
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[[분류:TQFT]]
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[[분류:migrate]]
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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==메타데이터==
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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===위키데이터===
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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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'}]
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