Volume of hyperbolic 3-manifolds

수학노트
http://bomber0.myid.net/ (토론)님의 2010년 3월 27일 (토) 20:36 판
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introduction
  • hyperbolic 3-manifold : figure 8 knot complement

 

 

 

volume 
  • 2.02988321281930725
    \(V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\)
    where D is Bloch-Wigner dilogarithm.
  • 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]

 

 

Chern-Simons invariant

 

 

 

Jones polynomial
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