Hyperbolic Knot Theory (Lecture - 2) by Abhijit Champanerkar

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 Hyperbolic Knot Theory (Lecture - 2)
0:00:28 In yesterday's talk
0:03:04 1 Structure of hyperbolic 3-manifolds
0:06:16 Margulis's thick-thin decomposition of non-compact hyperbolic
0:09:12 Hyperbolic Knots & links
0:11:49 (2, q)-torus knot diagrams:
0:13:52 Ideal triangulations
0:18:53 Idea : Give a hyperbolic structure-on inti(m ) by making each tet in I into a hyp. ideal tet
0:21:21 Faces : all ideal triangles in H2 are isometrics - gluing along faces is ok
0:22:15 Edges
0:26:10 Edge gluing equations
0:29:41 Vertices
0:34:57 Completeness equations
0:36:00 Theorem
0:40:50 Geometric complexity on Knots
0:41:42 Hyperbolic knot census
0:44:20 Checkerboard polyhedral & Example
0:49:04 Example
0:55:58 SnapPy
0:57:46 Hyperbolic geometry of virtual knots and links
1:00:51 Theorem - Colin Adams + REU students, Purcell- Howie
1:02:16 Q&A