"Volume of hyperbolic 3-manifolds"의 두 판 사이의 차이
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| − | + | ==introduction== | |
* hyperbolic 3-manifold<br> | * hyperbolic 3-manifold<br> | ||
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| − | + | ==volume of figure eight knot complement== | |
* http://mathworld.wolfram.com/FigureEightKnot.html<br> | * http://mathworld.wolfram.com/FigureEightKnot.html<br> | ||
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| − | + | ==history== | |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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| − | + | ==related items== | |
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| − | + | ==encyclopedia== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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| − | + | ==books== | |
| 103번째 줄: | 103번째 줄: | ||
| − | + | ==expositions== | |
* Steven Finch, Volumes of Hyperbolic 3-Manifolds, September 5, 2004 http://algo.inria.fr/csolve/hyp.pdf<br> | * Steven Finch, Volumes of Hyperbolic 3-Manifolds, September 5, 2004 http://algo.inria.fr/csolve/hyp.pdf<br> | ||
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| − | + | ==articles== | |
* Alexander Goncharov, [http://www.jstor.org/stable/2646189 Volumes of Hyperbolic Manifolds and Mixed Tate Motives], 1999 | * Alexander Goncharov, [http://www.jstor.org/stable/2646189 Volumes of Hyperbolic Manifolds and Mixed Tate Motives], 1999 | ||
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| − | + | ==question and answers(Math Overflow)== | |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
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| − | + | ==blogs== | |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
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| − | + | ==experts on the field== | |
* http://arxiv.org/ | * http://arxiv.org/ | ||
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| − | + | ==links== | |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
2012년 10월 28일 (일) 18:15 판
introduction
- hyperbolic 3-manifold
- three simple hyperbolic knots
- \(4_{1}\) figure 8 knot
- \(5_{2}\)
- \(6_{1}\), \(6_{1}\), \(6_{1}\)
- \(4_{1}\) figure 8 knot
volume of figure eight knot complement
- 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
- 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
- \(V(4_{1})=2.029883212819\cdots\)
- \(V(5_{2})=2.82812208\cdots\)
- \(V(6_{1})=3.163963228\cdots\)
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=
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
- 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