On the Khovanov and knot Floer homologies of quasi-alternating links

Abstract

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.

arXiv comment19 pages, 13 figures; minor revisions; to appear in Proceedings of the 14th Gokova Geometry / Topology Conference

References in corpus (6)

• Floer homology and knot complements
• Heegaard Floer homology and alternating knots
• Knot Floer homology detects fibred knots
• The support of the Khovanov's invariants for alternating knots
• Remarks on definition of Khovanov homology
• Knot Floer homology and integer surgeries

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