We give a purely combinatorial construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel features of the theory include the introduction of 'enhanced' foam facets which fix sign issues associated with the original matrix factorization formulation and the use of skew Howe duality to show that (enhanced) closed foams can be evaluated in a completely combinatorial manner. The latter answers a question posed in [40].

We show that Khovanov homology (and its sl 3 variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum slm via categorical skew Howe duality. Utilizing Cautis-Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones-Wenzl projectors and their sl 3 analogs purely from the higher representation theory of categorified quantum groups. In the sl 2 case, this work reveals the importance of a modified class of foams introduced by Christian Blanchet which in turn suggest a similar modified version of the sl 3 foam category introduced here. Contents

Abstract. We introduce an sln homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl 2 foams and categorified quantum gl m , via classical skew Howe duality. This framework then extends to give our annular sln link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sln sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sln, which in the n = 2 case recovers a result of Grigsby-Licata-Wehrli.

Abstract. We define and study the category of symmetric sl 2 -webs. This category is a combinatorial description of the category of all finite dimensional quantum sl 2 -modules. Explicitly, we show that (the additive closure of) the symmetric sl 2 -spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.

Abstract. We generalize results of Lee, Gornik and Wu on the structure of deformed colored sl N link homologies to the case of non-generic deformations. To this end, we use foam technology to give a completely combinatorial construction of Wu's deformed colored sl N link homologies. By studying the underlying deformed higher representation theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.