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Khovanov homology, sutured Floer homology and annular links

J Elisenda Grigsby and Stephan M Wehrli

Algebraic & Geometric Topology 10 (2010) 2009–2039

In [arXiv:0706.0741], Lawrence Roberts, extending the work of Ozsváth and Szabó in [Adv. Math 194 (2005) 1-33], showed how to associate to a link L in the complement of a fixed unknotB S3 a spectral sequence whose E2 term is the Khovanov homology of a link in a thickened annulus defined by Asaeda, Przytycki and Sikora in [Algebr. Geom. Topol. 4 (2004) 1177-1210], and whose E term is the knot Floer homology of the preimage of B inside the double-branched cover of L.

In [Adv. Math. 223 (2010) 2114-2165], we extended the aforementioned Ozsváth–Szabó paper in a different direction, constructing for each knot K S3 and each n +, a spectral sequence from Khovanov’s categorification of the reduced, n–colored Jones polynomial to the sutured Floer homology of a reduced n–cable of K. In the present work, we reinterpret Roberts’ result in the language of Juhasz’s sutured Floer homology [Algebr. Geom. Topol. 6 (2006) 1429–1457] and show that the spectral sequence of [Adv. Math. 223 (2010) 2114-2165] is a direct summand of the spectral sequence of Roberts’ paper.

Heegaard Floer homology, Khovanov homology, link invariants, branched covers
Mathematical Subject Classification 2000
Primary: 57M12, 57M27
Secondary: 57R58, 81R50
Received: 20 August 2009
Accepted: 20 November 2009
Published: 30 September 2010
J Elisenda Grigsby
Mathematics Department
Boston College
301 Carney Hall
Chestnut Hill MA 02467
Stephan M Wehrli
Mathematics Department
Syracuse University
215 Carnegie
Syracuse NY 13244