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A combinatorial approach to functorial quantum slk knot invariants
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 6, December 2009
- pp. 1679-1713
- 10.1353/ajm.0.0082
- Article
- Additional Information
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.