"Volume of hyperbolic 3-manifolds"의 두 판 사이의 차이

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5번째 줄: 5번째 줄:
 
** <math>4_{1}</math> figure 8 knot<br>
 
** <math>4_{1}</math> figure 8 knot<br>
 
** <math>5_{2}</math><br>
 
** <math>5_{2}</math><br>
−
** <math>6_{1}</math><br>
+
** <math>6_{1}</math>, <math>6_{1}</math>, <math>6_{1}</math><br>
  
 
 
 
 
19번째 줄: 19번째 줄:
 
*  obtained by glyeing two copies of ideal tetrahedra<br>
 
*  obtained by glyeing two copies of ideal tetrahedra<br>
 
*  thus the volume is given by<br><math>6\Lambda(\pi/3)</math> where [http://pythagoras0.springnote.com/pages/4630891 로바체프스키 함수]<br>
 
*  thus the volume is given by<br><math>6\Lambda(\pi/3)</math> where [http://pythagoras0.springnote.com/pages/4630891 로바체프스키 함수]<br>
−
*  2.02988321281930725<br><math>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</math><br> where D is [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm].<br>
+
*  2.02988321281930725<br><math>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</math><br> where D is [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm].<br>
 
**  this number is twice of [[Gieseking's constant]]<br>
 
**  this number is twice of [[Gieseking's constant]]<br>
 
*  what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>? numrically 1.285190955484149<br>
 
*  what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>? numrically 1.285190955484149<br>
31번째 줄: 31번째 줄:
 
<h5>other examples</h5>
 
<h5>other examples</h5>
  
 +
* <math>V(4_{1})=2.029883212819\cdots</math>
 
* <math>V(5_{2})=2.82812208\cdots</math>
 
* <math>V(5_{2})=2.82812208\cdots</math>
 
* <math>V(6_{1})=3.163963228\cdots</math>
 
* <math>V(6_{1})=3.163963228\cdots</math>

2010년 7월 30일 (금) 22:32 판

introduction
  • hyperbolic 3-manifold
  • three simple hyperbolic knots
    • \(4_{1}\) figure 8 knot
    • \(5_{2}\)
    • \(6_{1}\), \(6_{1}\), \(6_{1}\)

 

 

 

volume of figure eight knot complement
  • obtained by glyeing 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.
  • 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

 

 

Jones polynomial
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