Tuesday Seminar on Topology

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

2010/07/27

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ayumu Inoue (Tokyo Institute of Technology)
Quandle homology and complex volume
(Joint work with Yuichi Kabaya) (JAPANESE)
[ Abstract ]
For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M.
In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold.
He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group.
To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations.
On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring.
It means that we can compute the complex volume combinatorially from a link diagram.