Knot invariants and their categorifications via Howe duality

Series: 
Geometry Topology Seminar
Monday, April 13, 2015 - 14:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
USC
,  
 It is a well understood story that one can extract linkinvariants associated to simple Lie algebras.  These invariants  arecalled Reshetikhin-Turaev invariants and the famous Jones polynomialis the simplest example.  Kauffman showed that the Jones polynomialcould be described very simply by replacing crossings in a knotdiagram by various smoothings.  In this talk we will explainCautis-Kamnitzer-Licata's  simple new approach to understanding theseinvariants using basic representation theory and the quantum Weylgroup action. Their approach is based on a version of Howe duality forexterior algebras called skew-Howe duality.  Even the graphical (orskein theory) description of these invariants can be recovered in anelementary way from this data.   The advantage of this approach isthat it suggests a `categorification' where knot homology theoriesarise in an elementary way from higher representation theory and thestructure of categorified quantum groups. Joint work with David Rose and Hoel Queffelec