The sln foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality

Abstract

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].

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

“…An interesting feature of this relationship is that the natural system of symmetries in L m U q .gl m / actually translates into the braiding on Rep.sl n / which is crucial for the definition of the knot polynomials. This relationship was particular exploited in Lauda, Queffelec and Rose [13] and Queffelec and Rose [17], which categorify the features of the above technique and allows Lauda, Queffelec and Rose to provide a way that homology theories defined by Khovanov [10] and Khovanov and Rozansky [11] arise from categorified quantum groups.…”

“…The intention of this work is to set a possible grounding for a categorification in the style of [13] and [17], or of Cautis, Kamnitzer and Licata [3] and Cautis [2] which would provide a homology theory categorifying the Alexander polynomial that arises from representation theory, rather than from the categorifications arising from Floer homology. There is ongoing research into the connection between knot Floer homology and quantum sl n homology (see, for instance, Manolescu [15]), and a quantum gl.1j1/ homology theory may give a helpful intermediary.…”

Functoriality of colored link homologies

“…Proposition The defining relations [, (3.8)–(3.20)] of the foam 2‐category considered by Queffelec–Rose hold in .…”

Categorification of the colored 𝔰𝔩3-invariant

“…(2) Extend this construction to the sl n -homology defined by Queffelec and Rose in [20]. (3) Define geometrically the category of framed webs and framed foams, find a complete list of Reidemeister moves.…”