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Tuesday Seminar on Topology
Seminar information archive ～04/25｜Next seminar｜Future seminars 04/26～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
2024/11/05
17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Jun Murakami (Waseda University)
On complexified tetrahedrons associated with the double twist knots and its application to the volume conjecture (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Jun Murakami (Waseda University)
On complexified tetrahedrons associated with the double twist knots and its application to the volume conjecture (JAPANESE)
[ Abstract ]
In this talk, I first would like to explain complexified tetrahedrons which subdivide the complement of a double twist knot, and then talk about their application to the volume conjecture. Complexified tetrahedrons are introduced by using the fundamental group, and they are deformations of the regular ideal octahedron which is a half of the complement of the Borromean rings. Such deformations correspond to the surgeries of the Borromean rings producing the double twist knots.
On the other hand, the colored Jones polynomials of the double twist knots are given by the quantum 6j symbol with some extra terms. We see the correspondence of the quantum 6j symbol and the volume of a complexified terahedron by using the Neumann-Zagier function, and we can apply such correspondence to prove the volume conjecture for the double twist knots. To do this, the ADO invariant is used instead of the colored Jones invariant. The l'Hopital's rule is applied to get the ADO invariant, and integral by parts solves the big cancellation problem. At the last, it is shown that the application of the saddle point method is not so hard for this case.
[ Reference URL ]In this talk, I first would like to explain complexified tetrahedrons which subdivide the complement of a double twist knot, and then talk about their application to the volume conjecture. Complexified tetrahedrons are introduced by using the fundamental group, and they are deformations of the regular ideal octahedron which is a half of the complement of the Borromean rings. Such deformations correspond to the surgeries of the Borromean rings producing the double twist knots.
On the other hand, the colored Jones polynomials of the double twist knots are given by the quantum 6j symbol with some extra terms. We see the correspondence of the quantum 6j symbol and the volume of a complexified terahedron by using the Neumann-Zagier function, and we can apply such correspondence to prove the volume conjecture for the double twist knots. To do this, the ADO invariant is used instead of the colored Jones invariant. The l'Hopital's rule is applied to get the ADO invariant, and integral by parts solves the big cancellation problem. At the last, it is shown that the application of the saddle point method is not so hard for this case.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
