A Tour of Skein Modules by Rhea Palak Bakshi

PROGRAM

KNOTS THROUGH WEB (ONLINE)

ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh

DATE & TIME: 24 August 2020 to 28 August 2020

VENUE: Online

Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through online lectures.

Knots are fundamental objects of study in low dimensional topology and appear in diverse areas of sciences. Knot theory has seen tremendous progress in recent years. The aim of this online program is to familiarise and enthuse younger researchers about the latest advances in the subject with a particular emphasis on computational aspects of (co)homological, combinatorial and polynomial invariants of knots.

The pedagogical talks will be delivered by the following well-known experts in the field:

(1) Abhijit Champanerkar (City University of New York, USA)
(2) Andrei Vesnin (Sobolev Institute of Mathematics, Novosibirsk, Russia)
(3) Jozef H. Przytycki (The George Washington University, USA)
(4) Louis H. Kauffman (the University of Illinois at Chicago, USA)
(5) Mohamed Elhamdadi (University of South Florida, USA)
(6) Rhea Palak Bakshi (The George Washington University, USA)
(7) Seiichi Kamada (Osaka University, Osaka, Japan)
(8) Valeriy Bardakov (Sobolev Institute of Mathematics, Novosibirsk, Russia)

The primary audience of the program will be Ph.D. students, post-doctoral fellows, and young faculty members working in low dimensional topology and adjoining areas.

0:00:00 A Tour of Skein Modules
0:01:07 What is a skein module?
0:01:12 Alexander polynomial (1928)
0:01:55 Alexander - Conway polynomial (1969) Jones polynomial (1984)
0:03:02 Skein modules are generalizations of the various polynomial link invariants in $3 to arbitrary 3- manifolds.
0:04:09 Why are skein modules important?
0:04:25 Algebraic Geometry
0:05:21 OUTLINE
0:06:05 Notation
0:07:00 The Signed Skein Modules:
0:08:26 Framing Version of the Signed Skein Module
0:10:10 The Framing Skein Module
0:11:33 Dirac trick for a knot illustrated using a light bulb
0:11:54 Light bulb trick for a link
0:13:00 The q-homology Skein Module
0:14:21 The Third Skein Module
0:16:30 The Kth Skein Module
0:17:48 OTHER SKEIN MODULES
0:19:27 The Kauffman Bracket Skein Module
0:21:45 Properties of Kauffman Bracket Skein Modules
0:24:44 Examples of Kauffman Bracket Skein Modules
0:33:25 The KBSM of the Connected Sums of Handlebodies
0:35:28 The Kauffman Bracket Skein Algebra
0:39:11 Examples of Kauffman Bracket Skein Algebras
0:39:16 Theorem (Bullock - Przytycki, 2000)
0:41:52 The Product-to-Sum Formula
0:44:04 The KBSA of the Four-Punctured Sphere
0:45:24 (Bullock - Przytycki, 2000):
0:48:27 Theorem (Thurston, 2015)
0:49:55 Example of the KBSA of a Non-Oriented Surface
0:51:03 Properties of Kauffman Bracket Skein Algebras
0:54:02 Connection to the SL(2,C) Variety
0:55:07 Theorem (Culler - Shalen, 1983)
0:55:56 Theorem (Bullock, 1997)