"Volume of hyperbolic 3-manifolds"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
| 1번째 줄: | 1번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">introduction</h5> |
* hyperbolic 3-manifold : figure 8 knot complement<br> | * hyperbolic 3-manifold : figure 8 knot complement<br> | ||
| 9번째 줄: | 9번째 줄: | ||
| − | <h5 style="line-height: 2em; margin | + | <h5 style="line-height: 2em; margin: 0px;">volume </h5> |
| − | + | * 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> | |
| − | |||
| − | |||
* 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> | ||
| 48번째 줄: | 46번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">history</h5> |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| 56번째 줄: | 54번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">related items</h5> |
| 62번째 줄: | 60번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5> |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 72번째 줄: | 70번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">books</h5> |
| 87번째 줄: | 85번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5> |
| − | |||
| − | |||
| + | * [[#]]<br> | ||
| + | ** D.J. Broadhurst, 1998<br> | ||
* [[2010년 books and articles|논문정리]] | * [[2010년 books and articles|논문정리]] | ||
* http://www.ams.org/mathscinet | * http://www.ams.org/mathscinet | ||
* http://www.zentralblatt-math.org/zmath/en/ | * http://www.zentralblatt-math.org/zmath/en/ | ||
* http://pythagoras0.springnote.com/ | * http://pythagoras0.springnote.com/ | ||
| − | * http://math.berkeley.edu/~reb/papers/index.html[http://www.ams.org/mathscinet ] | + | * [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html][http://www.ams.org/mathscinet ] |
* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q= | * 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://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/ | + | * http://dx.doi.org/10.1007/s100529900935 |
| 104번째 줄: | 102번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">question and answers(Math Overflow)</h5> |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
| 113번째 줄: | 111번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs</h5> |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
| 123번째 줄: | 121번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">experts on the field</h5> |
* http://arxiv.org/ | * http://arxiv.org/ | ||
| 131번째 줄: | 129번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">links</h5> |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내] | * [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내] | ||
| − | * [http://www.research.att.com/ | + | * [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences] |
* http://functions.wolfram.com/ | * http://functions.wolfram.com/ | ||
| − | * | + | * |
2010년 3월 27일 (토) 21:36 판
introduction
- hyperbolic 3-manifold : figure 8 knot complement
volume
- 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. - 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]
Chern-Simons invariant
Jones polynomial
links
history
encyclopedia
- http://ko.wikipedia.org/wiki/
- 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
- #
- D.J. Broadhurst, 1998
- D.J. Broadhurst, 1998
- 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