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

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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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<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==
  
 
* volume of knot complements
 
* volume of knot complements
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==maps between threefolds</h5>
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==maps between threefolds==
  
 
* maps between aspherical 3 manifolds
 
* maps between aspherical 3 manifolds
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<h5 style="line-height: 2em; margin: 0px;">Volume of knot complement</h5>
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<h5 style="line-height: 2em; margin: 0px;">Volume of knot complement==
  
 
#  KnotData[]<br> KnotData["FigureEight", "HyperbolicVolume"]<br> N[%, 20]<br>
 
#  KnotData[]<br> KnotData["FigureEight", "HyperbolicVolume"]<br> N[%, 20]<br>
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<h5 style="line-height: 2em; margin: 0px;">a problem</h5>
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<h5 style="line-height: 2em; margin: 0px;">a problem==
  
 
*  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>
 
*  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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<h5 style="line-height: 2em; margin: 0px;">Reshetikihn, Turaev</h5>
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<h5 style="line-height: 2em; margin: 0px;">Reshetikihn, Turaev==
  
 
 
 
 
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==software</h5>
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==software==
  
 
* [http://www.geometrygames.org/SnapPea/ snappea]
 
* [http://www.geometrygames.org/SnapPea/ snappea]
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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>
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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==
  
 
* 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[[4667393|4667393]]</h5>
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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[[4667393|4667393]]==
  
 
* [[quantum dilogarithm]]<br>
 
* [[quantum dilogarithm]]<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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<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==
  
 
* http://en.wikipedia.org/wiki/Quantum_invariant<br>
 
* http://en.wikipedia.org/wiki/Quantum_invariant<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%;">books</h5>
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* http://www.worldscibooks.com/mathematics/4746.html<br>
 
* http://www.worldscibooks.com/mathematics/4746.html<br>
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<h5 style="line-height: 2em; margin: 0px;">expositions</h5>
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<h5 style="line-height: 2em; margin: 0px;">expositions==
  
 
*  Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier<br>
 
*  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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<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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<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==
  
 
* [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
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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)==
  
 
* http://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?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%;">blogs</h5>
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*  구글 블로그 검색<br>
 
*  구글 블로그 검색<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%;">experts on the field</h5>
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* http://arxiv.org/
 
* http://arxiv.org/
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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==
  
 
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]

2012년 10월 28일 (일) 15:56 판

introduction==
  • volume of knot complements
  • Chern-Simons invariant of manifolds
  • Turaev-Viro invariant (related to 6j symbols)
    • 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)
   

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==
  1. KnotData[]
    KnotData["FigureEight", "HyperbolicVolume"]
    N[%, 20]
  • Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
  • Bloch-Wigner dilogarithm is involved
   
a problem==
  • Prove
    \(\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))\)
  • a log tangent integral
   
Reshetikihn, Turaev==    

software

 

 

history==      

하위페이지

 

 

related items4667393==    
encyclopedia==    
books== 4909919      
expositions==    
articles==    
question and answers(Math Overflow)==    
blogs==    
experts on the field==    
links==
원본 주소 "https://wiki.mathnt.net/index.php?title=3-manifolds_and_their_invariants&oldid=43567"