Unraveling knots - four perspectives | Prof. Abhijit Champanerkar (CUNY) | Part 2

Speaker: Prof. Abhijit Champanerkar (College of Staten Island and The Graduate Center, The City University of New York)

Abstract: A knot is an embedding of the circle in the 3-space and is usually represented by a knot diagram. Knots appear in the artwork and symbolism of many cultures. Mathematical study of knots began around the late 18th century by Vandermonde and mid 19th century by Gauss. Motivated by physics, the celebrated Scottish physicist Peter Tait created the first knot tables. Knot theory has come a long way from its interesting origins. In this series of two talks we will present four major perspectives on knots - diagrammatic, topological, geometric and quantum. We will study how these idea play out in knot theory, see interesting theorems, and explore interactions of different fields in the context of knots. We will see generalizations of classical knot theory, recent developments and also visit some open problems in knot theory.

#MTTS #CurryLeaf #Abhijit_Champanerkar

Time Stamp

00:00:00 Welcome and Introduction to Prof. A V Jayanthan (Session Chair)
00:01:44 Words from Session Chair: Prof. A V Jayanthan
00:03:23 Introductory words from Session Chair: Prof. Abhijit Champanerkar
00:04:50 Outline of the talks
00:05:08 Outline of Today's talk(Talk 2)
00:05:25 What is a Topology?
00:08:06 Surfaces aka 2-dimensional manifolds
00:09:46 Knot complement
00:15:26 Fundamental Group
00:18:28 Alexander polynomial (1928)
00:24:04 Computing Alexander polynomial
00:26:12 Seifert surfaces
00:29:54 Seifert genus
00:31:42 Seifert matrix
00:36:42 Determinant, signature and Alexander polynomial
00:38:12 Beautiful pictures
00:38:59 Euclid of Alexandria, Father of Geometry
00:39:43 Euclid's Postulates
00:40:40 Problem with the parallel postulate
00:41:47 What if the Parallel Postulate wasn't true?
00:42:40 Hyperbolic Plane: Upper half-plane model
00:44:17 Hyperbolic geometry in art
00:45:45 2-dimensional geometries
00:46:15 Geometry on Surfaces
00:47:11 Three dimensional hyperbolic geometry
00:49:19 Hyperbolic building blocks
00:50:44 Hyperbolic knots
00:53:27 Knot invariants from geometry
00:55:48 Computing with hyperbolic knots - SnapPy
00:56:33 Hyperbolic Knot tabulation
01:01:33 Vol-Det Conjecture
01:03:36 The Volume conjecture
01:08:46 Generalisation 1-Knot Homologies
01:11:28 Generalisation 2- Virtual Knots
01:14:22 References
01:15:21 Questions and Answers
01:36:12 Vote of thanks

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Credits:
Thumbnail by: Nandana P Jayesh
Session Chair: Prof. A V Jayanthan, IIT Madras
Hosts: Nandana P Jayesh, Chaitanya Seth
Vote of thanks: Dr Soumya Dey
Timestamps by: Tamoghna Kar
Overall management: Dr Soumya Dey, CMI Post Doc. Fellow
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