A Combinatorial Approach To Functorial Quantum Sl(K) Knot Invariants

Mazorchuk, V. and Stroppel, C. (2009) A Combinatorial Approach To Functorial Quantum Sl(K) Knot Invariants. American Journal of Mathematics, 131(6), pp. 1679-1713.

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 essentially combinatorial arguments for the main results of Sussan. The justification that our combinatorial arguments and steps are correct uses non-conibinatorial 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 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 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 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Stroppel, Dr Catharina
• Authors: Mazorchuk, V., and Stroppel, C.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: American Journal of Mathematics
• Publisher: John Hopkins University Press
• ISSN: 0002-9327