| The thesis studies sl(n) and HOMFLY-PT link homology. We begin by constructing a version of sl(n)-link homology, which assigns the U(n)-equivariant cohomology of CP n-1 to the unknot. This theory specializes to the Khovanonv-Rozansky sl( n)-homology and we are motivated by the "universal" rank two Frobenius extension studied by M. Khovanov in (20) for sl(2)-homology. This framework allows one to work with graded, rather than filtered, objects and should prove useful in investigating structural properties of the sl(n)-homology theories.;We proceed by using the diagrammatic calculus for Soergel bimodules, developed by B. Elias and M. Khovanov in (1), to prove that Rouquier complexes, and ultimately HOMFLY-PT link homology, is functorial. Upon doing so we are able to explicitly write the chain map generators of the movie moves and compute over the integers. This is joint work with B. Elias.;In suite, we take the above diagrammatic calculus and construct and integral version of HOMFLY-PT link homology, which we also extend to an integral version of sl(n)-homology with the aid Rasmussen's specral sequences between the two (33). We reprove invariance under the Reidemeister moves in this context and highlight the computational power of the calculus at hand.;The last part of the thesis concerns an example. We show that for a particular class of tangles the sl(n)-link homology is entirely "local," i.e. has no "thick" edges, and its homology depends only on the underlying Frobenius structure of the algebra assigned to the unknot. |
