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Mathematics > Quantum Algebra

arXiv:0709.1971 (math)
[Submitted on 12 Sep 2007]

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

Authors:Volodymyr Mazorchuk, Catharina Stroppel
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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.
Comments: 29 pages, including several pictures
Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)
MSC classes: 17B10, 57M27, 05E10,81R50
Cite as: arXiv:0709.1971 [math.QA]
  (or arXiv:0709.1971v1 [math.QA] for this version)
  https://doi.org/10.48550/arXiv.0709.1971
Journal reference: American Journal of Mathematics 2009,

Submission history

From: Catharina Stroppel [view email]
[v1] Wed, 12 Sep 2007 23:46:32 UTC (169 KB)
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