##### Department of Mathematics,

University of California San Diego

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### Topology Seminar

## Qingtao Chen

## Recent development of Volume Conjecture of Kashaev, Reshetikhin-Turaev and Turaev-Viro invariants

##### Abstract:

In the late 1980's, after Jones' define his polynomial, there is a revolution in this area, followed by Witten's reinterpreting Jones polynomial by using Chern-Simons theory and predicting new quantum invariants. Finally Reshetikhin-Turaev was the first one to define a mathematically rigorous theory of such complex-valued invariant for closed 3-manifolds. More importantly, Reshetikhin-Turaev define their invariant not only at roots of unity $q(1)$ originally considered by Witten but also at other roots of unity. Later Turaev-Viro defined a real valued invariants for closed 3-manifolds by triangulation both at $q(1)$ and other roots of unity. In 1997, Kashaev discover his invariants of hyperbolic knots will become exponentially large as $N->infinity$ and he further conjectured that the growth rates corresponds to hyperbolic Volume of complement of that knot in $S^3$. In 2001, H. Murakami-J. Murakami extend Kashaev's Volume Conjecture from hyperbolic knots to all knots and hyperbolic volume to simplicial volume by using colored Jones polynomials. For many years, Witten-Reshetikhin-Turaev invariants evaluated at $q(1)$ was considered to be only polynomial growth and its asymptotic expansion is called WAE Conjecture (Witten's Asymptotic Expansion). Last year, in a joint work with T. Yang, we first define a real valued Turaev-Viro type invariant for 3-manifold with boundary by using ideal triangulation. Then we discovered this Turaev-Viro type invariant and Reshetikhin-Turaev invariant evaluated at other roots of unity (especially at $q(2)$) will have exponentially large phenomenon and the the growth rates corresponds to Volume of 3-manifold with boundary and Volume of closed 3-manifold respectively. Thankful to the new tool developed by Ohtsuki recently, asymptotic expansion of Kashaev invariants (including Volume Conjecture) up to 7 crossing has been solved. This new tool can also be used to attack my volume Conjecture with Tian Yang. I will give a brief introduction for all these new developments. Finally we expect Reshetikhin-Turaev at roots of unity other than $q(1)$ could have a different Geometry/Physics interpretation than original Chern-Simons theory given by Witten in 1989.

Organizer: Justin Roberts

### October 21, 2016

### 3:00 PM

### AP&M 7218

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