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

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.