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Abstract

This paper contains a categorification of the ${\frak sl}(k)$ link invariant using parabolic singular blocks of category ${\cal{O}}$. Our approach is intended to be as elementary as possible, providing essentially combinatorial arguments for the main results of Sussan. The justification that our combinatorial arguments and steps are correct uses non-combinatorial geometric and representation theoretic results (e.g., the Kazhdan-Lusztig and Soergel's theorems). We take these results as granted and use them like axioms (called {\it Facts\/} in the text).

We first construct an exact functor valued invariant of webs or special'' trivalent graphs labelled with $1, 2, k-1, k$ satisfying the MOY relations. Afterwards we extend it to the ${\frak sl}(k)$-invariant of links by passing to the derived categories. The approach of Khovanov using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the ${\frak sl}(k)$-homology, based on foams and the Kapustin-Li formula, from Mackaay, Stosic, and Vaz. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.

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