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

2011년 1월 6일 (목) 11:23 판

JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
- cutting M into
  - Seifert fibered pieces ~ non hyperbolic pieces
  - atoroidal pieces ~ hyperbolic pieces
Thurston's geometrization
- S^3, E\times S^2, Sol
- E^3, E\times H^2, SL_2
- H^3, Nil
Knot complements
- K\

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)

Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
Bloch-Wigner dilogarithm is involved

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

[[4909919|]]

Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links
- J.M. Borwein, D.J. Broadhurst, 1998
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

@@ 3번째 줄: / 3번째 줄: @@
 * maps between aspherical 3 manifolds
 * aspherical threefolds = second and higher homotopy groups vanish
-*  backgrounds<br>
-** geometrization
 *  JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition<br>
-** cutting M into Seifert fibered PCs
+**  cutting M into<br>
+*** Seifert fibered pieces ~ non hyperbolic pieces
+*** atoroidal pieces ~ hyperbolic pieces
+*  Thurston's geometrization<br>
+** S^3, E\times S^2, Sol
+** E^3, E\times H^2, SL_2
+** H^3, Nil
+*  Knot complements<br>
+** K\