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A COMBINATORIAL APPROACH TO FUNCTORIAL QUANTUM sl(k) KNOT INVARIANTS
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 131, no 6, 1679-1713 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2009. Vol. 131, no 6, 1679-1713 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-127385ISI: 000272641900005OAI: oai:DiVA.org:uu-127385DiVA: diva2:329777
Available from: 2010-07-13 Created: 2010-07-13 Last updated: 2017-12-12Bibliographically approved

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