"Figure eight knot"의 두 판 사이의 차이
imported>Pythagoras0 (새 문서: ==introduction== Answered by [http://mathoverflow.net/users/1345/agol Agol]: The [http://en.wikipedia.org/wiki/Figure_eight_knot_%28mathematics%29 figure eight knot] (complement) is t...) |
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 4개는 보이지 않습니다) | |||
| 4번째 줄: | 4번째 줄: | ||
==volume== | ==volume== | ||
| − | * http://mathworld.wolfram.com/FigureEightKnot.html | + | * http://mathworld.wolfram.com/FigureEightKnot.html |
| − | * obtained by glueing two copies of ideal tetrahedra | + | * obtained by glueing two copies of ideal tetrahedra |
| − | * thus the volume is given by | + | * thus the volume is given by:<math>6\Lambda(\pi/3)</math> where [http://pythagoras0.springnote.com/pages/4630891 로바체프스키 함수] |
| − | * 2.02988321281930725 | + | * 2.02988321281930725:<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> where D is [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm]. |
| − | ** this number is twice of [[Gieseking's constant]] | + | ** this number is twice of [[Gieseking's constant]] |
| − | * what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>? numrically 1.285190955484149 | + | * what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>? numrically 1.285190955484149 |
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| + | ==memo== | ||
| + | * [http://www.mtholyoke.edu/acad/math/figure_eight.html Figure Eight Knot Sculpture] | ||
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==computational resource== | ==computational resource== | ||
| − | * https://docs.google.com/file/d/ | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxZnNVZ0NLVXU0Z00/edit |
| + | [[분류:migrate]] | ||
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| + | == 메타데이터 == | ||
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| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q930352 Q930352] | ||
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| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q930352 Q930352] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'figure'}, {'OP': '*'}, {'LOWER': 'eight'}, {'LEMMA': 'knot'}] | ||
2021년 2월 17일 (수) 01:28 기준 최신판
introduction
Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.
volume
- 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
- what is \(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)\)? numrically 1.285190955484149
memo
computational resource
메타데이터
위키데이터
- ID : Q930352
메타데이터
위키데이터
- ID : Q930352
Spacy 패턴 목록
- [{'LOWER': 'figure'}, {'OP': '*'}, {'LOWER': 'eight'}, {'LEMMA': 'knot'}]