Volume Conjecture in Tokyo

## Speakers

• Jinseok Cho (Busan)
• Stavros Garoufalidis (Gatech)
• Sergei Gukov (Caltech)
• Kazuhiro Hikami (Kyushu)
• Rinat Kashaev (Geneve)
• Thang T.Q. Le (Gatech)
• Hitoshi Murakami (Tohoku)
• Jun Murakami (Waseda)
• Christian Zickert (UMD)

## Program

Reception starts at 13:00 on 22nd.
22nd (Wed.)23rd (Thu.) 24th (Fri.)
10:00--11:00 //////////////////////////Kashaev (talk 1) Le (talk1)
11:30--12:30 //////////////////////////Kashaev (talk 2) Le (talk 2)
lunch break
14:00--15:00 ZickertGukov Hikami
15:30--16:30 Hitoshi MurakamiJun Murakami Cho
17:00--18:00 GaroufalidisFree Discussion

## Titles and Abstracts

• Jinseok Cho (Busan)
• Title : The optimistic limit of colored Jones polynomial
Abstract : We survey the optimistic limit of the colored Jones polynomial and its application to the Ptolemy variety.
• Stavros Garoufalidis (Gatech)
• Title : The volume conjecture for state integrals
Abstract : I will discuss the volume conjecture for state integrals of cusped hyperbolic 3-manifolds. Joint work with Joergen Andersen and Rinat Kashaev.
• Sergei Gukov (Caltech)
• Title: "Zed-hat"
Abstract: The goal of the talk will be to introduce a function that answers a question in topology, can be computed via methods more common in the theory of dynamical systems, and in the end turns out to enjoy beautiful modular properties of the type first observed by Ramanujan.
• Kazuhiro Hikami (Kyushu)
• Title: On Kauffman bracket skein algebra and DAHA
Abstract:I will talk about a quantization of character variety, and discuss relationship between skein algebra on surface and DAHA.
• Rinat Kashaev (Geneve)
• Title: The Volume conjecture for knots in arbitrary 3-manifolds
Abstract: In the first lecture, I will recall the construction of quantum dilogarithmic invariants of links in arbitrary (closed compact oriented) 3-manifolds which, in the case of 3-sphere, coincide with the N-colored Jones polynomial at an N-th root of unity (the result of H. Murakami and J. Murakami). In the case of 3-manifolds different from 3-sphere, the volume conjecture has not been tested yet, partly because of technical difficulties of actual calculations.
• Thang T.Q. Le (Gatech)
• Title of talk 1: On growth of torsion homology of 3-manifolds
Abstract: We discuss some results/conjectures on the growth of torsion homology of 3-manifolds and their relations to work of Kojima and McShane on bounds of hyperbolic volumes and work of McMullen, Liu, and Hadari on the homological stretch factor of pseudo-Anosov maps.

Title of talk 2: The skein algebra of surfaces and hyperbolic TQFT
Abstract: The skein algebra of surface has close relations to the character variety and the quantum Teichmuller space; it serves as a bridge between quantum topology and classical topology. We will present some recent results in representations of the skein algebra and show their potential applications in hyperbolic TQFTs.
• Hitoshi Murakami (Tohoku)
• Title: The volume conjecture for cable knots
Abstract: I will talk about relations of the asymptotic behaviors of the colored Jones polynomials of cable knots to the Chern-Simons invariants and the Reidemeister torsions. A part of the talk is a joint work of A. Tran.
• Jun Murakami (Waseda)
• Title: Volume conjecture for the logarithmic invariant
Abstract: The logarithmic invariant of knot is introduced by using the symmetric linear function of the small quantum group, which is a version of the quantum group $\mathcal U_q(sl_2)$ at a root of unity. Symmetric linear function means a linear function whose value for a product $xy$ is equal to opposite product $yx$, and there is a symmetric linear function other than a trace since the small quantum group is not a semisimple algebra. This invariant is redefined and extended to links by using the modified traces of the modules of the small quantum group by Beliakova, Blanchet and Geer, where the modified trace is the symmetric linear functions satisfying a special property defined for the modules in the tensor category of the representations of the small quantum group. In this talk, I would like to explain the logarithmic invariant and the volume conjecture of this invariant. As the original volume conjecture, this conjecture gives a relation between the logarithmic invariant and the hyperbolic volume of the corresponding 3-manifold, and it also seems to hold for links and knotted graphs in a 3-manifold. The logarithmic invariant looks like a colored version of Kashaev's invariant, so it may have some relation to the Kashaev's original construction of his invariant.
• Christian Zickert (UMD)
• Title: Ptolemy varieties and Bloch invariants for higher dimensional manifolds
Abstract: We discuss how the theory of shape and Ptolemy coordinates on representation varieties extends to higher dimensional manifolds.

## Acknowledgements

This workshop is partially supported by JSPS KAKENHI Grants:17K18728 (J. Murakami), 16K05145 (T. Tanaka), 16K05162 (T. Tsukamoto), 17K05240 (Y. Yamaguchi).

## Organizers

Yuanyuan Bao (U-Tokyo), Noboru Ito (U-Tokyo), Yoshiyuki Yokota (Tokyo Metropolitan)