# Skein Algebra and related topics

# June 14 - 20, 2023

University of Science Vietnam National University Ho Chi Minh City (VNUHCM-US), 227 Nguyen Van Cu District 5 Ho Chi Minh City; and Premier Residences Phu Quoc Emerald Bay (Phu Quoc Island).

## Mini-courses

Wednesday June 14 to Saturday June 17, 2023. Room F207, University of Science Vietnam National University Ho Chi Minh City (VNUHCM-US), 227 Nguyen Van Cu District 5 Ho Chi Minh City.

- Andrew Kricker (Nanyang Technological University):
, 4 hour mini-course**Introduction to 3-D index** - Thang T. Q. Le (Lê Tự Quốc Thắng) (Georgia Institute of Technology):
, 4 hour mini-course**Skein Algebra of Surfaces and Quantum Traces** - Adam Sikora (State University of New York at Buffalo):
, 2 hour mini-course**Introduction to Skein modules**

## Talks

Sunday June 18 to Tuesday June 20, 2023, to be held at Premier Residences Phu Quoc Emerald Bay (Phu Quoc Island)

## Organizers

- Thang T. Q. Le, email: This email address is being protected from spambots. You need JavaScript enabled to view it.
- Huỳnh Quang Vũ (local organizer) email: This email address is being protected from spambots. You need JavaScript enabled to view it.

## Preliminary program

June 14

09:00-10:00 Adam Sikora: **Introduction to 3-dimensional topology and to skein modules**

10:30-11:30 Adam Sikora: **Skein modules and Representations of Fundamental Groups**

June 15

08:00-09:00 Thang Le: **Skein modules/algebras and quantum traces****I**

09:15-10:15 Andrew Kricker: *Introduction to 3-D index I*

10:45-11:45 Thang Le: *Skein modules/algebras and quantum traces II*

June 16

08:00-09:00 Andrew Kricker: *Introduction to 3-D index II*

09:15-10:15 Thang Le: *Skein modules/algebras and quantum traces III*

10:45-11:45 Andrew Kricker: *Introduction to 3-D index III*

14:00-15:00 Adam Sikora: **Computing skein modules over fields**

15:30-16:30 Zhe Sun: **Asymmetric intersection number and Fock-Goncharov duality**

June 17

08:00-09:00 Thang Le: *Skein modules/algebras and quantum traces IV*

09:15-10:15 Andrew Kricker: *Introduction to 3-D index IV*

10:45-11:45 Tsukasa Ishibashi:** Moduli space of decorated G-local systems and skein algebras**

June 18 – from June 18, the talks will be held at Premier Residences Phu Quoc Emerald Bay (Phu Quoc Island).

09:00-10:00 Jeffrey Andre Weenink: *GL(n) skein theory from U(1) and SL(n)*

10:45-11:45 Wataru Yuasa: *Skein and cluster algebras of unpunctured surfaces for sp_4*

June 19

09:00-10:00 Zhihao Wang : *Kauffman bracket intertwiners and the volume conjecture*

10:45-11:45 Hiroaki Karuo: *Quantum duality maps and skein algebras*

June 20

09:00-10:00 Thang Le: *TBA*

## Titles and Adstracts

**Mini-courses**

**Andrew Kricker **(Nanyang Technological University):** Introduction to 3-D index**

Abstract: The 3D-index is a fascinating construction which associates to an ideal triangulation of a cusped 3-manifold and a choice of first homology class on its boundary a power series in a formal variable.

It was discovered in 2011 in the context of supersymmetric gauge theories by the theoretical physicists Dimofte, Gaiotto and Gukov and was formulated mathematically by Garoufalidis in 2016.

Although the physics predicts it is a topological invariant, the 3D-index is not defined on every ideal triangulation, so understanding the topological meaning of this construction is quite a subtle and surprisingly rich problem.

One of the reasons the theory is so fascinating is because it is deeply connected to the rich combinatorics associated to ideal triangulations, such as the gluing matrices of Neumann and Zagier and the theory of normal surfaces.

In these lectures we’ll aim to introduce some of the following:

- An introduction to ideal triangulations and some of their associated combinatorics.
- Various versions of the definition of the 3D-index of an ideal triangulation.
- An introduction to normal surfaces and the connection to the 3D-index.
- The state-integral version of 3D-index due to Garoufalidis and Kashaev.

The presentation will be based on the papers:

- “1-efficient triangulations and the index of a cusped hyperbolic 3-manifold”, by S. Garoufalidis, C. Hodgson, H. Rubinstein, and H. Segerman. Geometry and Topology, 19 (5) 2619-2689, 2015.
- “The 3D-index and normal surfaces”, by S. Garoufalidis, C. Hodgson, N. Hoffman, and H. Rubinstein. Illinois Journal of Math, 60 (1) 289-352, 2016.
- “A meromorphic extension of the 3D-index”, by S. Garoufalidis and R. Kashaev. Research in the mathematical sciences, 6:8, 2019.

**Thang Le** (Georgia Institute of Technology):** Skein modules/algebras and quantum traces**

Abstract: Knot theory plays an important role in topology and has interesting relations to many remote branches of mathematics and physics, like number theory and non-commutative algebras.

The skein module of a 3-manifold is defined using links in the 3-manifold with relations coming from identities of invariants of links in R^3. When the 3-manifold is a thickened surface the skein module has a natural algebra structure, and it quantizes the character variety of the surface. We first give a short introduction to the skein module/algebra theory and sketch a connection between the skein algebra theory and hyperbolic geometry/cluster algebra theory via (higher) Teichmuller theory. We will explain how to construct the quantum traces, both the X- and A-versions, for SL_2 and SL_n skein algebras.

**Talks**

**Tsukasa Ishibashi** (Tohoku University): *Moduli space of decorated G-local systems and skein algebras *

Abstract: The moduli space of decorated (twisted) G-local systems on a marked surface, originally introduced by Fock–Goncharov, is known to have a natural cluster K_2 structure. In particular, it admits a quantization via the framework of quantum cluster algebras, due to Berenstein—Zelevinsky and Goncharov—Shen.

In this talk, I will explain its (in general conjectural) relation to the skein algebras: we have two generating sets of the function ring of the moduli space, and the quantum lift of their relations would lead to an isomorphism between the reduced stated skein algebra and the boundary-localized Muller-type skein algebra.

This talk is based on a joint work with Hironori Oya and Linhui Shen, and a joint work with Wataru Yuasa.

**Hiroaki Karuo **(Gakushuin University):** Quantum duality maps and skein algebras**

Abstract: In the context of quantum Teichm\"uller theory, conjectured were the existence of quantizations of trace functions of Teichm\"uller spaces and some expected properties. A construction of such quantizations, called the quantum duality maps, was given for punctured surfaces by Allegretti--Kim and for marked disks by Allegretti. Allegretti--Kim's strategy was based on skein algebras and Bonahon--Wong's quantum trace maps. One of the expected properties is the positivity of structure constants of the images of the quantum duality maps, so-called the positivity conjecture. Although Mandel--Qin showed the coincidence of theta bases and their quantum duality maps and the positivity conjecture as a corollary, it is still important to understand the positivity with explicit formulas of structure constants in terms of skein algebras.

In the talk, we try to understand the positivity of structure constants using reduced stated skein algebras. This is a joint work with Ishibashi Tsukasa (Tohoku University).

**Jeffrey Andre Weenink** (Nanyang Technological University): *GL(n) skein theory from U(1) and SL(n)*

**Wataru Yuasa **(Kyoto University): *Skein and cluster algebras of unpunctured surfaces for sp_4*

Abstract: We introduce a skein algebra consisting of $\mathfrak{sp}_4$-webs on a marked surface without punctures and its $\mathbb{Z}_q$-subalgebra called the ``$\mathbb{Z}_q$-form''. We prove that the boundary-localization of the $\mathbb{Z}_q$-form is included in a quantum cluster algebra that quantizes the function ring of the moduli space of all decorated twisted $Sp_4$-local systems on the surface. In this talk, we explain the construction of the inclusion and a conjectural characterization of cluster variables using $\mathfrak{sp}_4$-webs. This talk is based on a joint work with Tsukasa Ishibashi

**Zhihao Wang** (Nanyang Technological University): *Kauffman bracket intertwiners and the volume conjecture*

Abstract: Bonahon-Wong-Yang introduced a new version of the volume conjecture by using the intertwiners between two isomorphic irreducible representations of the skein algebra. The intertwiners are built from surface diffeomorphisms; they made the volume conjecture when diffeomorphisms are pseudo-Anosov. We calculated all the intertwiners for the closed torus, producing some interesting results. Moreover we considered the periodic diffeomorphisms, and introduced the corresponding conjecture. For some special cases we made precise calculations for intertwiners and proved our conjecture.

**Zhe Sun** (University of Science and Technology of China ): **Asymmetric intersection number and Fock-Goncharov duality**

Fock and Goncharov introduced a pair of mirror moduli spaces associated to G and G^L which generalized the Teichmüller space and the decorated Teichmüller space, and they proposed a duality: the canonical basis of the regular function ring of one space X is parameterized by the tropical integral points of its mirror X^V. In this talk, I will explain my joint work with Linhui Shen and Daping Weng for SL3 (in progress for SLn), where we introduce the topological asymmetric intersection numbers between webs on the surfaces to provide the duality pairings and the map from webs to tropical points. We prove that the map is the same as the previous one obtained by Douglas and myself. We relate the cluster algebra and skein algebra by this intersection number and prove the mutation equivariance, where the flip equivariance is a consequence. If time permitted, I will also explain my joint work with Tsukasa Ishibashi and Wataru Yuasa for the Sp4 case.

## Registration

Please let us know if you are coming by filling the registration form.