Knot Floer homology in cyclic branched covers

Grigsby, J Elisenda

doi:10.2140/agt.2006.6.1355

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Knot Floer homology in cyclic branched covers

J Elisenda Grigsby

Algebraic & Geometric Topology 6 (2006) 1355–1398

DOI: 10.2140/agt.2006.6.1355

arXiv: math.GT/0507498

Abstract

In this paper, we introduce a sequence of invariants of a knot $K$ in $S^{3}$ : the knot Floer homology groups $\hat{H F K} (Σ^{m} (K); \tilde{K}, i)$ of the preimage of $K$ in the $m$ –fold cyclic branched cover over $K$ . We exhibit $\hat{H F K} (Σ^{m} (K); \tilde{K}, i)$ as the categorification of a well-defined multiple of the Turaev torsion of $Σ^{m} (K) - \tilde{K}$ in the case where $Σ^{m} (K)$ is a rational homology sphere. In addition, when $K$ is a two-bridge knot, we prove that $\hat{H F K} (Σ^{2} (K); \tilde{K}, s_{0}) ≅ \hat{H F K} (S^{3}; K)$ for $s_{0}$ the spin Spin $^{c}$ structure on $Σ^{2} (K)$ . We conclude with a calculation involving two knots with identical $\hat{H F K} (S^{3}; K, i)$ for which $\hat{H F K} (Σ^{2} (K); \tilde{K}, i)$ differ as $ℤ_{2}$ –graded groups.

Keywords

Heegaard Floer homology, branched covers

Mathematical Subject Classification 2000

Primary: 57R58, 57M27

Secondary: 57M05

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Publication

Received: 9 September 2005

Accepted: 10 June 2006

Published: 25 September 2006

Authors

J Elisenda Grigsby
Evans Hall University of California, Berkeley Berkeley, CA 94720 USA

Volume 6, issue 3 (2006)