Khovanov Homology and Virtual Knot Cobordism by Louis H. Kauffman

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 Khovanov Homology and Virtual Knot Cobordism
0:00:14 Khovanov Homology
0:00:22 Cubism
0:02:40 Use enhanced states by labeling each loop with +I or -1.
0:04:32 Enhanced States
0:07:25 The differential should increase the homological grading i by I and leave fixed the quantum grading j.
0:08:58 The boundary is a sum of partial differentials corresponding to resmoothings on the states.
0:10:33 Cubism
0:14:36 Checking Order Compatibility
0:21:38 Cubism Again
0:24:27 Dror's Canopoly
0:24:37 We examine this question as though we had not seen the Frobenius algebra
0:29:05 Complexes for Second Reidemeister Move
0:31:04 Figure 15: Four-Tube Relation From Homotopy
0:32:15 Schematic Four-Tube Relation
0:32:38 The dot can be taken to represent an algebra element
0:33:58 The Tube Relation implies the Four Tube Relation.
0:34:05 Coproduct via the Tube-Relation
0:34:14 From 4Tu to Frobenius Algebra
0:36:42 Algebra from 4Tu - Guaranteed to Produce Link Homology
0:40:48 Grading
0:41:04 Khovanov Homology for Virtual Links (our latest evolution of Manturov's approach)
0:42:36 But composing along the opposite sides we see
0:43:30 Solution over the integers by counting transport through virtual crossings
0:46:15 Virtual Knot Cobordism
0:47:00 Figure 17: Virtual Stevedore is Slice
0:48:11 Virtual Stevedore is not classical.
0:48:29 Vertical Mirror Image
0:48:40 Connected Sum with the Vertical Mirror Image
0:48:47 We say that K is concordant to K
0:49:07 Spanning Surfaces for Knots and Virtual Knots.
0:50:30 Figure 20: Virtual Cobordism Seifert Surface
0:51:30 Heather Dye, Aaron Kaestner and LK, prove the following generalization of Rasmussen's Theorem,
0:52:12 This Gauss code schema shows how to produce infinitely
0:52:36 Affine Index Polynomial (See LK and Folwazcny and variants from Henrich, Cheng, Dye,...)
0:57:05 Concordances are Composed of Elementary Concordances (Cancellation of Saddle and Max or Min)
0:57:42 Theorem. P_K and PF_K are concordance invariants.
1:01:36 Thank you for your attention!