3-manifolds and their invariants

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

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
  $$ \begin{align} \frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt & =\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7})) \\ & = \frac{2}{\sqrt{7}}(Cl(2\pi /7)+Cl(4\pi/7)-Cl(6\pi/7)) \end{align} $$
  
• a log tangent integral
  

 

invariants

• 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)
• Chern-Simons invariant
• Kashaev's volume conjecture
• Triangulations and the Bloch group
• Volume of hyperbolic threefolds and L-values and volume of knot complements
• Number fields and threefolds
• Reidemeister torsion

 

 

software

• snappea
• snap
• http://www.math.utk.edu/~morwen/knotscape.html

 

 

history

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

 

related items

• Topological quantum field theory(TQFT)
• quantum dilogarithm
• threefolds and their invariants
• Chern-Simons invariant
• Gieseking's constant
• mathematics of x^3-x+1=0
• triangulations and Bloch group
• volume of hyperbolic threefolds and L-values

 

encyclopedia

• http://en.wikipedia.org/wiki/Quantum_invariant
  
• http://ko.wikipedia.org/wiki/[1]
• http://en.wikipedia.org/wiki/

 

 

books

• http://www.worldscibooks.com/mathematics/4746.html
  
• 2010년 books and articles
  

 

 

 

expositions

• Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier
  
• Christian Blanchet, Vladimir Turaev Quantum Invariants of 3-manifolds
  

 

 

articles

• 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:10.1142/S0217751X96001905. http://arxiv.org/abs/hep-th/9505102.
• Three-manifolds and the Temperley-Lieb algebra W. B. R. Lickorish, 1991
• Hyperbolic manifolds and special values of Dedekind zeta-functions Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
  

• http://dx.doi.org/10.1063/1.3085764

 

 

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/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences