"Volume of hyperbolic 3-manifolds"의 두 판 사이의 차이
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| 2번째 줄: | 2번째 줄: | ||
* hyperbolic 3-manifold : figure 8 knot complement<br> | * hyperbolic 3-manifold : figure 8 knot complement<br> | ||
| + | * <math>5_{2}</math><br> | ||
| + | * <math>6_{1}</math><br> | ||
| 25번째 줄: | 27번째 줄: | ||
| − | <h5> | + | <h5>other examples</h5> |
| + | |||
| + | <math>V(5_{2})=2.82812208\cdots</math> | ||
| + | |||
| + | <math>V(6_{1})=3.163963228\cdots</math> | ||
| 99번째 줄: | 105번째 줄: | ||
* [http://www.jstor.org/stable/2646189 Volumes of Hyperbolic Manifolds and Mixed Tate Motives]<br> | * [http://www.jstor.org/stable/2646189 Volumes of Hyperbolic Manifolds and Mixed Tate Motives]<br> | ||
** Alexander Goncharov, 1999 | ** Alexander Goncharov, 1999 | ||
| + | * [http://www.jstor.org/stable/2001854 Hyperbolic invariants of knots and links]<br> | ||
| + | ** Adams, C., Hildebrand, M. and Weeks, J., Trans. Amer.Math. Soc. 1 (1991), 1–56. | ||
* [http://www.springerlink.com/content/v36272439g3g5006/ Hyperbolic manifolds and special values of Dedekind zeta-functions]<br> | * [http://www.springerlink.com/content/v36272439g3g5006/ Hyperbolic manifolds and special values of Dedekind zeta-functions]<br> | ||
** Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월 | ** Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월 | ||
2010년 7월 30일 (금) 21:57 판
introduction
- hyperbolic 3-manifold : figure 8 knot complement
- \(5_{2}\)
- \(6_{1}\)
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
\(V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\)
where D is Bloch-Wigner dilogarithm.
- this number is twice of Gieseking's constant
- this number is twice of Gieseking's constant
- what is \(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)\)? numrically 1.285190955484149
- 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
\(V(5_{2})=2.82812208\cdots\)
\(V(6_{1})=3.163963228\cdots\)
Chern-Simons invariant
Jones polynomial
links
history
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
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=
[[4909919|]]
articles
- Volumes of Hyperbolic Manifolds and Mixed Tate Motives
- Alexander Goncharov, 1999
- Hyperbolic invariants of knots and links
- Adams, C., Hildebrand, M. and Weeks, J., Trans. Amer.Math. Soc. 1 (1991), 1–56.
- Hyperbolic manifolds and special values of Dedekind zeta-functions
- Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
- Commensurability classes and volumes of hyperbolic 3-manifolds
- A. Borel, Ann. Sc. Norm. Super. Pisa8, 1–33 (1981)
- 논문정리
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html[1]
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- http://dx.doi.org/10.1007/s100529900935
question and answers(Math Overflow)
blogs
experts on the field