"3-manifolds and their invariants"의 두 판 사이의 차이
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| − | + | ==introduction== | |
* volume of knot complements | * volume of knot complements | ||
| 28번째 줄: | 28번째 줄: | ||
| − | + | ==Volume of knot complement== | |
# KnotData[]<br> KnotData["FigureEight", "HyperbolicVolume"]<br> N[%, 20]<br> | # KnotData[]<br> KnotData["FigureEight", "HyperbolicVolume"]<br> N[%, 20]<br> | ||
| 39번째 줄: | 39번째 줄: | ||
| − | + | ==a problem== | |
* Prove<br><math>\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}))</math><br> | * Prove<br><math>\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}))</math><br> | ||
| 48번째 줄: | 48번째 줄: | ||
| − | + | ==Reshetikihn, Turaev== | |
| 64번째 줄: | 64번째 줄: | ||
| − | + | ==history== | |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| 87번째 줄: | 87번째 줄: | ||
| − | + | ==related items[[4667393|4667393]]== | |
* [[quantum dilogarithm]]<br> | * [[quantum dilogarithm]]<br> | ||
| 95번째 줄: | 95번째 줄: | ||
| − | + | ==encyclopedia== | |
* http://en.wikipedia.org/wiki/Quantum_invariant<br> | * http://en.wikipedia.org/wiki/Quantum_invariant<br> | ||
| 106번째 줄: | 106번째 줄: | ||
| − | + | ==books== | |
* http://www.worldscibooks.com/mathematics/4746.html<br> | * http://www.worldscibooks.com/mathematics/4746.html<br> | ||
| 119번째 줄: | 119번째 줄: | ||
| − | + | ==expositions== | |
* Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier<br> | * Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier<br> | ||
| 128번째 줄: | 128번째 줄: | ||
| − | + | ==articles== | |
* [http://arxiv.org/abs/hep-th/9811173 Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links] J.M. Borwein, D.J. Broadhurst, 1998 | * [http://arxiv.org/abs/hep-th/9811173 Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links] J.M. Borwein, D.J. Broadhurst, 1998 | ||
| 141번째 줄: | 141번째 줄: | ||
| − | + | ==question and answers(Math Overflow)== | |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
| 150번째 줄: | 150번째 줄: | ||
| − | + | ==blogs== | |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
| 160번째 줄: | 160번째 줄: | ||
| − | + | ==experts on the field== | |
* http://arxiv.org/ | * http://arxiv.org/ | ||
| 168번째 줄: | 168번째 줄: | ||
| − | + | ==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) 17:05 판
introduction
- volume of knot complements
- Chern-Simons invariant of manifolds
- 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)
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
- cutting M into
- 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
- 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
\(\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))\) - a log tangent integral
Reshetikihn, Turaev
software
history
하위페이지
encyclopedia
- http://en.wikipedia.org/wiki/Quantum_invariant
- http://ko.wikipedia.org/wiki/[1]
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
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
question and answers(Math Overflow)
blogs
experts on the field