A combinatorial approach to functorial quantum sl(k) knot invariants

Abstract

This paper contains a categorification of the sl(k) link invariant using parabolic singular blocks of category O. Our approach is intended to be as elementary as possible, providing combinatorial proofs of the main results of Sussan. 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 sl(k)-invariant of links by passing to the derived categories. The approach using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the sl(k)-homology, based on foams and the Kapustin-Lie formula. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.

29 pages, including several pictures

References in corpus (2)

• The universal sl3-link homology
• Quadratic duals, Koszul dual functors, and applications

Cited by in corpus (4)

• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• Knot invariants and higher representation theory II: the categorification of quantum knot invariants
• Khovanov-Rozansky homology via Cohen-Macaulay approximations and Soergel bimodules
• A categorification of the quantum sl(N)-link polynomials using foams