Categorification and applications in topology and representation theory

TL;DR

This thesis advances the categorification of link invariants by extending Bar-Natan's cobordism approach to virtual links and introduces a new algebraic structure, the sl3 web algebra, connecting topology with representation theory.

Contribution

It extends topological link homologies to virtual links and introduces the sl3 web algebra, linking categorification with algebraic and geometric structures.

Findings

Extended Bar-Natan's categorification to virtual links

Constructed a computable topological complex for virtual link homologies

Defined the sl3 web algebra and proved its properties, including Morita equivalence and relation to cohomology rings

Abstract

This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows, over rings of characteristic 2, extensions with no classical analogon, e.g. Bar-Natan's Z/2Z-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICA based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology. In the second part of this thesis (which is based on joint work with Mackaay and Pan) we use Kuperberg's sl3 webs and Khovanov's sl3 foams to define a new algebra K_S, which we call the sl3 web algebra. It is the sl3 analogue of Khovanov's arc algebra H_n. We prove that K_S is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that K_S is Morita equivalent to a certain cyclotomic KLR-algebra. This allows us to determine the split Grothendieck group K_0(K_S), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that K_S is a graded cellular algebra.