# Workshop on Volume Conjecture and Quantum Topology

### -- a new perscective on low-dimensional geometry --

September 6 (Tue) to 9 (Fri), 2016

at Room 54-204, 54th building,
Nishi-Waseda Campus, Waseda University
Tokyo, JAPAN
(30 seconds from Nishiwaseda subway Station)

### Program

September 6 (Tuesday)

9:20 ~     Registration

10:00 ~ 11:00 Alexander Kolpakov

11:15 ~ 12:15 Shunsuke Tsuji

15:45 ~ 16:45 Masahito Yamazaki

17:00 ~ 18:00 Don Zagier

18:00 ~ Welcome reception (free)

at Takeuchi Lounge (2F of 55S building)

September 7 (Wednesday)

10:00 ~ 11:00 Delphine Moussard

11:15 ~ 12:15 Yoshiyuki Yokota

14:30 ~ 15:30 Dongmin Gang

16:00 ~ 17:00 Stavros Garoufaridis

September 8 (Thursday)

10:00 ~ 11:00 Roland van der Veen

11:15 ~ 12:15 Hitoshi Murakami

14:30 ~ 15:30 Rei Inoue

16:00 ~ 17:00 Rinat Kashaev

18:00 ~ Dinner (5,000 yen, 3,000 yen for students)

September 9 (Friday)

10:00 ~ 11:00 Qingtao Chen

11:15 ~ 12:15 Toshie Takata

14:30 ~ 15:30 Kazuo Habiro

16:00 ~ 17:00 Bertrand Patureau-Mirand

### A State integral model for $$SL(2)$$ Chern-Simons theory on closed hyperbolic 3-manifolds

#### Donming Gang (IPMU)

I will give a state-integral model for closed 3-manifolds obtained by incorporating Dehn's filling into known state-integral models on knot complements. After analyzing perturbative/non-perturbative aspects of the state-integral model with several explicit examples, I will explain some potential applications of the state-integral in both physics and mathematics.

### Can you count genus 2 surfaces?

#### Stavros Garoufalidis (Georgia Tech)

I will give a classical, modern, post-modern and futuristic introduction to genus 2 surfaces and normal surfaces in 3-manifolds, motivated by the physics of the $$3D$$index. Joint work with australian (and semi-australian) collaborators Hodgson, Hoffmann, Rubinstein and Segerman.

### Extended Kontsevich integral for bottom tangles in handlebodies

#### Kazuo Habiro (RIMS, Kyoto University)

Using the Kontsevich integral, we define a functor from the category of bottom tangles in handlebodies to a category of chord diagrams. This functor can be thought of as a (partial) refinement of the LMO functor on Lagrangian cobordisms. This is joint work with Gwenael Massuyeau.

### Cluster braiding operator and Kashaev's R-matrix

#### Rei Inoue (Chiba University)

I review an application of cluster algebra to study knot invariants, based on joint work with Kazuhiro Hikami. Especially we relate the braiding operator given by cluster mutations to Kashaev's R-matrix.In three-dimensional hyperbolic geometry, a cluster mutation corresponds to an ideal tetrahedron, and cluster variables and coefficients respectively correspond to Zickert's edgeparameters and the modulus of the tetrahedron. We define the octahedral R-operator composed of four mutations, and state a conjecture on the complex volume of knot complements in $$S^3$$ in terms of the cluster algebra. Further, by using Fock-Goncharov's $$q$$-deformation of cluster algebra, we construct a braiding operator in terms of quantum dilogarithm functions. In a limit that $$q$$ goes to a root of unity, the braiding operator reduces to Kashaev's R-matrix.

### Invariants of surface bundles in quantum Teichmüller theory

#### Rinat Kashaev (Université de Genéve)

Quantum Teichmüller theory gives rise unitary projective representations of the mapping class groups of punctured surfaces in infinite dimensional Hilbert spaces. An application of that theory in 3-manifold topology is the construction of invariants of surface bundles by taking the traces of the associated quantum operators. A priori, the trace of a unitary operator in an infinite dimensional Hilbert space is not well defined but the volume conjecture for the Teichmüller TQFT implies that those traces should be finite at least in the case of hyperbolic surface bundles. I will discuss the cases of finite cyclic coverings of the trefoil and figure-eight knot complements.

### Around volumes of Coxeter polytopes, scissors congruence and related conjectures

#### Alexander Kolpakov (University of Toronto)

In this talk I will first outline the recent results of the joint work with Jun Murakami (Waseda University) on the Volume Conjecture for hyperbolic polytopes. I shall discuss a number of computer experiments and make some plausible conjectures based on them.

Second, I will switch to the spherical case and discuss some work in progress with Sinai Robins (ICERM & University of Sao Paolo) concerning volumes of spherical simplices and some distant connections with Hilbert's scissors congruence problem and the Rational Simplex Conjecture by Cheeger and Simons.

### Finite type invariants of knots in rational homology 3-spheres

#### Delphine Moussard (Université de Bourgogne)

We introduce a theory of finite type invariants for null-homologous knots in rational homology 3-spheres which can be thought as a rational homology version of the theory studied by Garoufalidis and Kricker for knots in integral homology 3-spheres. We study the graded space associated to this theory, whose dual is the space of rational valued finite type invariants, graded by the degree. We express this graded space as a quotient of a diagram space and give a complete combinatorial description in the case of knots whose Alexander polynomial has degree at most two. The goal of this work is to provide a comparison of the Kricker lift of the Kontsevich integral and of the Lescop invariant constructed by means of equivariant intersections in configuration spaces, the equivalence of which has been conjectured by Lescop.

### The colored Jones polynomial and representations of a knot group

#### Hitoshi Murakami (Tohoku University)

As a generalization of the volume conjecture, I will talk about a relation of the colored Jones polynomial of a knot with representations of the fundamental group of its complement to $$SL(2;{\mathbb C})$$.

### A survey on the asymptotic expansion of the Kashaev invariant of hyperbolic knots with up to 7 crossings

#### Tomotada Ohtsuki (RIMS, Kyoto University)

The asymptotic expansion of the Kashaev invariant of hyperbolic knots is a refinement of the volume conjecture. This asymptotic expansion for some hyperbolic knots can be calculated by using the Poisson summation formula and the saddle point method. By definition, the Kashaev invariant of a knot is presented by a certain sum. By using the Poisson summation formula, we can rewrite this sum by an integral. Further, by using the saddle point method, we can calculate the asymptotic expansion of this integral. In this talk, I explain a survey on this asymptotic expansion for hyperbolic knots with up to 7 crossings.

#### Bertrand Patureau-Miland (Université de Bretagne-Sud)

Several topological invariants of links in $$S^3$$ can be built from quantum group $${\mathcal U}_q sl(2)$$. Among them, the "semi-simple" colored Jones polynomials and the "non semi-simple" Akutsu-Degushi-Ohtsuki invariants of links in $$S^3$$, their intersection being the Kashaev-Murakami-Murakami invariant subject to the volume conjecture.

In the semi-simple setting, the link invariants are the basis of the construction of the Witten-Reshetikhin-Turaev invariant of 3-manifolds and of a 2+1-TQFT. I will explain how similar constructions are possible starting from the ADO invariants. This lead to a kind of TQFT for $$C^*$$-tangles in 3-manifolds. From the underlying 3-manifold invariant we can form a generalize volume conjecture.

This is a common work with Christian Blanchet, François Costantino and Nathan Geer.

### On the asymptotic expansions of the Kashaev invariant of some hyperbolic knots with 8 crossings

#### Toshie Takata (Kyushu University)

We give presentations of the asymptotic expansions of the Kashaev invariant of some hyperbolic knots with 8 crossings. We show the volume conjecture for these knots.

### Completed Kauffman bracket skein algebras and an invariant for integral homology 3-spheres

#### Shunsuke Tsuji (University of Tokyo)

Using an explicit formula for the action of the Dehn twist along a simple closed curve on the completed Kauffman bracket skein module of the surface, we introduce an embedding of the Torelli group into the completed skein algebra. This embedding and a Heegaard splitting enable us to construct an invariant for an integral homology sphere which is an element of $$\mathbb{Q} [[A+1]]$$. This invariant induces a finite type invariant of order $$n$$ which is an element of $$\mathbb{Q}[[A+1]]/((A+1)^n$$.

### Normally ordered exponentials in quantum doubles, from Alexander to Jones

#### Roland van der Veen (Leiden University)

We introduce a family of quantum doubles $$SM_n$$ of the two-dimensional Lie algebra and discuss the universal invariants they give rise to. The algebras $$SM_n$$ may be viewed as approximations to the standard quantum group $${\mathcal U}_q(gl_2)$$. This suggests a close relationship with the colored Jones polynomials, especially their loop expansion. We focus on the doubles $$SM_0$$ and $$SM_1$$ and explain how our technique of normally ordered exponentials allows one to calculate their invariants effectively. $$SM_0$$ gives rise to a version of the Alexander polynomial. The $$SM_1$$ invariant is more powerful and may deserve a closer look. Joint work with Dror Bar-Natan.

### Quantum hyperbolic geometry and cluster $$3d$$ $$N=2$$ theories

#### Masahito Yamazaki (IPMU)

We discuss a curious relation between $$3d$$ quantum hyperbolic geometry
and a class of $$3d$$ $$N=2$$ supersymmetric field theories as prescribed by quantum
cluster algebras.

### On a non-positively curved cubing of the exterior of an alternating knot

#### Yoshiyuki Yokota (Tokyo Metropolitan University)

The purpose of this talk is to show that, for an alternating knot $$K$$, the edges of the ideal triangulation of the complement of $$K$$ related to Kashaev's invariant are essential. The basic tool is a non-positively curved cubing of the exterior of $$K$$ which is directly obtained from this triangulation.

### Quantum invariants, $$q$$-series, and Bloch groups

#### Don Zagier (MPIM Bonn)

This workshop is supported by JSPS KAKENHI Grant Numbers 16H03931, 16H06336, 25287014.

Organizer: Jun Murakami (Waseda University)
email: murakami_at_waseda.jp (replace _at_ by @)