On Khovanov's cobordism theory for SU3 knot homology

Abstract

We reconsider the 3 link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local 2 theory of [2]. We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's 3 spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory. Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the 3 homology of the (2,n) torus knots. Show more