Khovanov homology, sutured Floer homology and annular links

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 unknot $B \subset S^{3}$ a spectral sequence whose $E^{2}$ 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 $E^{\infty}$ 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 $K \subset S^{3}$ and each $n \in ℤ_{+}$ , 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.

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Keywords

Heegaard Floer homology, Khovanov homology, link invariants, branched covers

Mathematical Subject Classification 2000

Primary: 57M12, 57M27

Secondary: 57R58, 81R50

References

Publication

Received: 20 August 2009

Accepted: 20 November 2009

Published: 30 September 2010

Authors

J Elisenda Grigsby
Mathematics Department Boston College 301 Carney Hall Chestnut Hill MA 02467
http://www2.bc.edu/~grigsbyj/
Stephan M Wehrli
Mathematics Department Syracuse University 215 Carnegie Syracuse NY 13244

Volume 10, issue 4 (2010)

J Elisenda Grigsby and Stephan M Wehrli

Abstract