Unraveling knots - four perspectives | Prof. Abhijit Champanerkar (CUNY) | Part 1

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 Dr Ajit Kumar (Session Chair)
00:02:39 Words from Session Chair: Dr Ajit Kumar
00:08:47 Introductory words from Session Chair: Prof. Abhijit Champanerkar
00:12:00 Unraveling knots - four perspectives
00:13:30 Outline of the talks
00:14:12 Outline of Today's talk(Talk 1)
00:15:17 What is a Knot?
00:18:06 Knot Diagrams
00:20:11 Careful of Wild knots
00:22:38 Equivalence of Knot Diagrams
00:26:13 Knots in art and culture
00:27:34 Knots and ether
00:29:57 New from old
00:37:20 Special families: Alternating knots
00:39:12 Special families: Torus knots
00:45:40 How to tell knots apart?
00:48:51 Crossing number
00:50:42 Knots with low crossing number
00:52:00 Knot tabulation
00:58:02 Tait Conjecture
01:02:01 Tricolorability
01:06:35 Knots and Graphs
01:12:10 Knot determinant
01:13:09 Jones Polynomial
01:16:46 Kauffman bracket polynomial
01:18:16 Effect under Riedemeister moves
01:22:23 Kauffman states
01:25:30 Spanning tree model of Jones polynomial
01:31:01 Trefoil skein tree and spanning trees
01:31:54 Tait Conjecture
01:32:56 Programs and databases of knot invariants
01:34:30 References
01:35:25 Image credits
01:35:50 Questions and Answers
01:48:53 Vote of thanks

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Credits:
Thumbnail by: Nandana P Jayesh
Session Chair: Prof. Ajit Kumar, ICT MUMBAI
Hosts: Tamoghna Kar, Nandana P Jayesh
Vote of thanks: Nandana P Jayesh
Timestamps by: Tamoghna Kar
Overall management: Dr Soumya Dey, CMI Post Doc. Fellow
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