"3-manifolds and their invariants"의 두 판 사이의 차이

2010년 3월 23일 (화) 19:56 판

volume of figure 8 knot complement
\(V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})\)
what is \(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)\)?

\(\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})\)

\(\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))\)

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 <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>
-*  volume of figure 8 know complement<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}})</math><br>
+*  volume of figure 8 knot complement<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}})</math><br>
 *  what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>?<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>
+* [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm]<br>
+복소이차수체의 [http://pythagoras0.springnote.com/pages/4533335 데데킨트 제타함수]
+<math>\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})</math>
+<math>\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))</math>