This is an informal and convivial space whose main goal is to learn and discuss about low-dimensional topology and related topics. People who are completely new to the subject are very welcome to join us. If you would like to receive information about our activities please write an email to

## Registration

If you participate in the seminar or if you want to receive information about the seminar (including Zoom link) please answer the survey here (even if you will not give a talk).

If you are willing to give the presentation, please choose all the chapters you want to present. If you have any questions and suggestions, please don't hesitate to contact us.

## Special Lectures

Speaker: Prof. Louis H. Kauffman (University of Illinois at Chicago)
Time: August 19 (Friday), 10:00 am (KST = BJT+1 = GMT+9 )
Venue: Zoom Meeting

Abstract: This talk will review the structure of the Temperley-Lieb Recoupling Theory and describe some of its applications, including the Fibonacci model for topological quantum computation, invariants of three manifolds and recent work by the speaker and Heather Dye and Eiji Ogasa on invariants of virtual knots and links and associated three manifolds.

### The geometry of $q$-$6j$-symbols

Speaker: Prof. Nicolai Reshetikhin
Time: May 2, 3-5 pm (KST = BJT+1)
Venue: Zoom Meeting

Abstract: $6j$-symbols, also known as Racah-Wigner coefficients, first appeared in representation theory of $SU(2)$, or, as it is called in physics, the theory of quantum angular momentum. They are given by special values of hypergeometric functions ${}_4F_3$ and have a geometric limit which relates them with the geometry of coadjoint orbits in the dual space to the Lie algebra $\mathfrak{su}(2)$.

Their $q$-analogs appear in representation theory of quantum universal enveloping algebra of $\mathfrak{sl}(2)$. They are given by special values of $q$-hypergeometric functions ${}_4\phi_3$ and are instrumental in topological quantum field theory and corresponding conformal field theories. They also have a geometric limit and are related to the geometry of $SU(2)$ conjugation orbits.

In this talk, I will start with the definition of $q$-$6j$ symbols, explain how to express colored Jones polynomials in terms of these objects. Then I will focus on the relation of $q$-$6j$ symbols with the Clebsch-Gordan integrable system for $\mathfrak{su}(2)$ and will compute their semiclassical asymptotic. At the end, I will review the results of Q. Chen, J. Murakami, T. Yang and others on the relation between this semiclassical asymptotic and the volume conjecture.

### Introduction to Topological Quantum Field Theories

Time: March 4, 9-10 am (KST = BJT+1)
Venue: Zoom Meeting

Abstract: The lecture will be an introduction to quantum topology for beginners (including undergraduates). I will start by a brief description of the Alexander and Jones polynomials of knots and links in the $3$-dimensional sphere. Then I will introduce the concept of an $n$-dimensional topological quantum field theory (TQFT) and discuss algebraic sources of such theories for $n=2$ and $n=3$. Preliminary knowledge of the following notions is required: a manifold of a given dimension $n=1,2,3, \ldots$ a manifold with boundary, a homeomorphism of manifolds, an algebra, a module over an algebra, a group, a category, a knot in the $3$-dimensional sphere, a link in the $3$-dimensional sphere, a braid on $n$ strings.

## Spring, 2022

During this season of the Learning seminar we will focus on the study of quantum invariants
of 3-manifolds by using Temperley-Lieb algebras. We will closely follow the book:

• Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds by Louis H. Kauffman and Sóstenes L. Lins)

Our main goal is understanding the definition of Turaev-Viro and Reshetikhin-Turaev
invariants and their explicit computation. In particular we will learn about:

• Kauffman bracket and Temperley-Lieb algebras.
• Triangulations, spines and Heegaard splittings of 3-manifolds.
• Turaev-Viro invariants.
• Reshetikhin-Turaev invariants.

Here is a more detailed introduction.

Starting from March, the learning seminar will take place every Monday, from 3pm to 5pm (KST=GMT+9), via Zoom

Meeting ID: 541 134 7674

Archive
Speaker Time Topic Archive
Vladimir Turaev Mar. 04 Introduction to Topological Quantum Field Theories Video
Anderson Vera Mar. 07 Bracket polynomial, Temperley-Lieb algebra Overview, Note, Video
Yifan Li Mar. 14 Jones-Wenzl projectors Overview, Note, Video
Minkyoung Song Mar. 21 The 3-vertices Note
Seonhwa Kim Mar. 28 Properties of projectors and 3-vertices Video, Note
Seongjeong Kim Apr. 04 θ -Evaluations Note, Video
Overview: Ch.3-Ch.6
MyeongSang Cho Apr. 11 Recoupling theory via Temperley-Lieb algebra Video
Eric Dolores Apr. 18 Chromatic evaluations and the tetrahedron I Note, Video
Eric Dolores Apr. 25 Chromatic evaluations and the tetrahedron II Note, Video
Nicolai Reshetikhin May 2 The geometry of $q$-$6j$-symbols Note, Video
Seonhwa Kim May 9 A $3$-manifold Invariant by state summation Note, Video
Yifan Li May 16 The Shadow World Note, Video
Eric Dolores May 23 The Witten-Reshetikhin-Turaev invariant I Video
Eric Dolores June 13 The Witten-Reshetikhin-Turaev invariant II Note, Video
Louis H. Kauffman Aug 19 Temperley-Lieb Recoupling Theory and Invariants of Classical and Virtual Knots and Links Note1, Note2, Video

## 2021

During this year we studied the theory of quantum and Vassiliev invariants of knots and their relations through the Kontsevich integral. Our main references were the books:

1. An introduction to Quantum and Vassiliev Knot invariants” by David M. Jackson and Iain Moffat.
2. Quantum Invariants. A study of knots, 3-manifolds and their sets” by Tomotada Ohstuki.
Archive
Speaker   Time Topic Archive
Seonhwa Kim (SNU) 13/07 Basic knot theory and some knot invariants
Minkyoung Song (IBS-CGP) 20/07 Braids and its R-matrix representations
Anderson Vera (Postech) 27/07 Knot invariants through R-matrix representations Video1, video2
Yifan Li (IBS-CGP) 03/08 Operator invariants of tangles Video, note
Yong-Geun Oh (IBS-CGP, Postech) 10/08 Ribbon Hopf algebras I
Yong-Geun Oh (IBS-CGP, Postech) 17/08 Ribbon Hopf algebras II
Anderson Vera (Postech) 24/08 Ribbon Hopf algebras III Video1, video2
Yifan Li (IBS-CGP) 31/08 Turaev-Reshetikhin invariants Video, note
Minkyoung Song (IBS-CGP) 06/09 Fundamentals of Vassiliev invariants Video
Seonhwa Kim (SNU) 13/09 Chord diagrams I
Seonhwa Kim (SNU) 27/09 Chord diagrams II
Yifan Li (IBS-CGP) 18/10 Jacobi diagrams I Video, note
Anderson Vera (Postech) 25/10 Jacobi diagrams II Video
Myeong-Sang Cho (Postech) 01/11 Lie algebra weight systems I Video
Myeong-Sang Cho (Postech) 08/11 Lie algebra weight systems II Video
Anderson Vera (Postech) 29/11 q-tangles and Jacobi diagrams on 1-manifols I
Anderson Vera (Postech) 06/12 q-tangles and Jacobi diagrams on 1-manifols II Video
Yifan Li (IBS-CGP) 13/10 Construction of Kontsevich Integral Video, note
Seonhwa Kim (SNU) 20/12 Universality properties of Kontsevich Invariants
Xinxing Tang (BIMSA) 21/12 The KZ equations and the Drinfel’d associator I Video
Xinxing Tang (BIMSA) 22/12 The KZ equations and the Drinfel’d associator II Video, note

News/Related

Calendar