A categorification of the quantum sl(N)-link polynomials using foams

TL;DR

This thesis develops a foam-based categorification of the sl(N)-link polynomial for N≥3, establishing functorial homology theories that relate to existing Khovanov-Rozansky homology and exploring torsion phenomena.

Contribution

It introduces a new foam-based framework for categorifying sl(N)-link polynomials, including a universal sl(3)-link homology and a rational theory for N≥4, connecting to known homologies.

Findings

The universal sl(3)-link homology is functorial up to scalars.

The foam-based theories are isomorphic to Khovanov-Rozansky homology.

The integral sl(N)-link homology for (2,m)-torus links contains N-torsion.

Abstract

In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for N\geq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show that tensoring it with Q yields a theory which is equivalent to the rational universal Khovanov-Rozansky sl(3)-link homology. For N\geq 4 we construct a rational theory categorifying the sl(N)-link polynomial using foams. Our theory is functorial, up to scalars, with respect to link cobordisms. To evaluate closed foams we use the Kapustin-Li formula. We show that for any link our homology is isomorphic to the Khovanov-Rozansky homology. We conjecture that the theory is integral and we compute the conjectured integral sl(N)-link homology for the (2,m)-torus links and show that it has torsion of order N.