Polynomial Invariants of Virtual Knots by Andrei Vesnin

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 Polynomial Invariants of Virtual Knots
0:01:51 The talk is based on the following papers:
0:02:40 Content
0:03:43 Virtual knot diagrams
0:05:04 Classical crossing sign
0:06:26 Labelling of arcs
0:08:27 Index value
0:10:07 n-dwrithe
0:12:03 Properties of the n-dwrithe
0:13:02 Flat virtual knots
0:14:24 Reversing orientation and mirror image
0:15:54 L-polynomials of a virtual knot diagram
0:16:04 Two types of smoothing
0:18:26 Definition of L-polynomials
0:21:34 Proof of Theorem 1. Move RI
0:25:52 Proof of Theorem 1. Move Rll
0:26:40 Proof of Theorem 1. Move Rlll
0:28:02 Proof of Theorem 1. Move SV
0:29:04 Thus, in Case (1) as well as in Case (2) we get
0:29:34 Example 1(a)
0:31:07 Example 1(b)
0:32:12 Example 1(c)
0:33:45 Example 1(d)
0:34:19 Example 1(e)
0:35:23 L-polynomials and Kauffman affine index polynomial (a)
0:37:08 L-polynomials and Kauffman affine index polynomial (b)
0:38:42 Application 1: Cosmetic crossing change conjecture
0:38:50 The case of classical knots
0:39:30 A Kirby's list problem on classical knots
0:40:47 A problem on virtual knots
0:42:07 F-polynomials of a virtual knot diagram
0:42:14 Definition of F-polynomials
0:44:18 F-polynomials are invariants of virtual knots
0:44:51 Example 3
0:45:58 Application 2: Mutation by positive involution
0:46:06 Conway mutation
0:47:14 Conway mutation and F-polynomials
0:47:38 Example 4
0:49:42 Tabulation of F-polynomials
0:50:56 Table of virtual knots (a-d)
0:52:01 Table of F-polynomials, part 1
0:53:55 Table of F-polynomials, part 5
0:54:09 Table of F-polynomials, part 6
0:54:12 Table of F-polynomials, part 7
0:54:50 Thank you!
0:54:55 Q&A