"Volume of hyperbolic 3-manifolds"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
(피타고라스님이 이 페이지를 개설하였습니다.)
 
1번째 줄: 1번째 줄:
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">introduction</h5>
  
 +
*  hyperbolic 3-manifold : figure 8 knot complement<br>
 +
 +
 
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">volume </h5>
 +
 +
 2.02988321281930725
 +
 +
* <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>
 +
 +
#  L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]<br> f[x_, y_] :=<br>  L[x] + L[1 - x*y] + L[y] + L[(1 - y)/(1 - x*y)] + L[(1 - x)/(1 - x*y)]<br> Print["five term relation"]<br> Table[f[i, j], {i, 0.1, 0.9, 0.1}, {j, 0.1, 0.9, 0.1}] // TableForm<br> N[3 L[Exp[2 I*Pi/3]], 20]<br> N[2 L[Exp[I*Pi/3]], 20]<br> N[3 (L[Exp[2 I*Pi/3]] - L[Exp[4 I*Pi/3]])/2, 20]<br> N[Pi^2*L[Exp[2 I*Pi/3]]/(3 Sqrt[3]), 20]<br>
 +
 +
 
 +
 +
 
 +
 +
<h5>Chern-Simons invariant</h5>
 +
 +
 
 +
 +
* [[Chern-Simons gauge theory and invariant|Chern-Simons theory]]
 +
 +
 
 +
 +
 
 +
 +
<h5>Jones polynomial</h5>
 +
 +
*  
 +
 +
 
 +
 +
 
 +
 +
<h5>links</h5>
 +
 +
* [http://pythagoras0.springnote.com/pages/5098745 매듭이론 (knot theory)]
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">history</h5>
 +
 +
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">related items</h5>
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">encyclopedia</h5>
 +
 +
* http://ko.wikipedia.org/wiki/
 +
* http://en.wikipedia.org/wiki/
 +
* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">books</h5>
 +
 +
 
 +
 +
* [[2010년 books and articles]]<br>
 +
* http://gigapedia.info/1/
 +
* http://gigapedia.info/1/
 +
* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 +
 +
[[4909919|]]
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">articles</h5>
 +
 +
 
 +
 +
* [[2010년 books and articles|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[http://www.ams.org/mathscinet ]
 +
* 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/
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">question and answers(Math Overflow)</h5>
 +
 +
* http://mathoverflow.net/search?q=
 +
* http://mathoverflow.net/search?q=
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">blogs</h5>
 +
 +
*  구글 블로그 검색<br>
 +
** http://blogsearch.google.com/blogsearch?q=
 +
** http://blogsearch.google.com/blogsearch?q=
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">experts on the field</h5>
 +
 +
* http://arxiv.org/
 +
 +
 
 +
 +
 
 +
 +
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">links</h5>
 +
 +
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 +
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 +
* [http://www.research.att.com/~njas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 +
* http://functions.wolfram.com/
 +
* <br>

2010년 3월 26일 (금) 08:01 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
  1. 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

 

 

related items

 

 

encyclopedia

 

 

books

 

[[4909919|]]

 

 

articles

 

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
원본 주소 "https://wiki.mathnt.net/index.php?title=Volume_of_hyperbolic_3-manifolds&oldid=37583"