Volume of hyperbolic 3-manifolds

introduction

• volume is an important invariant of hyperbolic 3-manifold
• big open problem Kashaev's volume conjecture
• three simple hyperbolic knots
  
  • \(4_{1}\) figure 8 knot
    
  • \(5_{2}\)
    
  • \(6_{1}\), \(6_{1}\), \(6_{1}\)
    

 

 

 

volume of figure eight knot complement

• http://mathworld.wolfram.com/FigureEightKnot.html
  

• obtained by glueing two copies of ideal tetrahedra
  
• thus the volume is given by
  \(6\Lambda(\pi/3)\) where 로바체프스키 함수
  
• 2.02988321281930725
  \(V(4_{1})=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\)
  where D is Bloch-Wigner dilogarithm.
  
  • this number is twice of Gieseking's constant
    
• what is \(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)\)? numrically 1.285190955484149
  

1. L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]
   f[x_, y_] :=
    L[x] + L[1 - x*y] + L[y] + L[(1 - y)/(1 - x*y)] + L[(1 - x)/(1 - x*y)]
   Print["five term relation"]
   Table[f[i, j], {i, 0.1, 0.9, 0.1}, {j, 0.1, 0.9, 0.1}] // TableForm
   N[3 L[Exp[2 I*Pi/3]], 20]
   N[2 L[Exp[I*Pi/3]], 20]
   N[3 (L[Exp[2 I*Pi/3]] - L[Exp[4 I*Pi/3]])/2, 20]
   N[Pi^2*L[Exp[2 I*Pi/3]]/(3 Sqrt[3]), 20]
   

 

 

other examples

• \(V(4_{1})=2.029883212819\cdots\)
• \(V(5_{2})=2.82812208\cdots\)
• \(V(6_{1})=3.163963228\cdots\)

 

 

Chern-Simons invariant

• Chern-Simons theory

 

 

•  

 

 

links

• 매듭이론 (knot theory)

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxX3ZsSC04OEUwU0k/edit

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
• http://en.wikipedia.org/wiki/

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

expositions

• Steven Finch, Volumes of Hyperbolic 3-Manifolds, September 5, 2004 http://algo.inria.fr/csolve/hyp.pdf
  

 

 

articles

• Alexander Goncharov, Volumes of Hyperbolic Manifolds and Mixed Tate Motives, 1999
• Gliozzi, F., and R. Tateo. 1995. Thermodynamic Bethe Ansatz and Threefold Triangulations. hep-th/9505102 (May 17). doi:doi:10.1142/S0217751X96001905. http://arxiv.org/abs/hep-th/9505102. 
• Adams, C., Hildebrand, M. and Weeks, J., Hyperbolic invariants of knots and links, Trans. Amer.Math. Soc. 1 (1991), 1–56.
• Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions,  Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
• A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds , Ann. Sc. Norm. Super. Pisa8, 1–33 (1981)
• 논문정리

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/