Institut de Géométrie, Algèbre et Topologie

Cours de IIIe cycle

Khovanov Homology

Paul Turner (Heriot-Watt and Fribourg)

Fridays, 10h15 to 12h

(dates: February 20, 27; March 6, 27; April 3, 24; May 1)

BCH 5112

 

 

Contents

Around 1998 Mikhail Khovanov invented a homology theory for knots and links which provided a novel way of viewing well-established quantum knot invariants (for example the Jones polynomial and the HOMFLY polynomial). Over the last few years this area has seen considerable development not only in knot theory and low- dimensional topology but also in representation theory, geometry and mathematical physics.

The aim of this course will be to introduce Khovanov's original homology theory for knots and links. Topics to be covered include: knot theory background; the Khovanov complex, its homology and elementary properties; Frobenius algebras and $1+1$-dimensional topological quantum field theories; invariance of Khovanov homology under Reidemeister moves; related theories including Lee theory; functoriality with respect to link cobordisms; tools from algebraic topology; the skein exact sequence; Lee's spectral sequence; Rasmussen's invariant and the slice genus; torsion; overview of Khovanov-Rozansky theory and Heegard-Floer knot homology; overview of some recent development

Prerequisites: Basic knowledge of knot theory, low-dimensional and algebraic topology