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Knot Floer homology in cyclic branched covers

J Elisenda Grigsby

Algebraic & Geometric Topology 6 (2006) 1355–1398

arXiv: math.GT/0507498

Abstract

In this paper, we introduce a sequence of invariants of a knot K in S3: the knot Floer homology groups HFK̂(Σm(K);K˜,i) of the preimage of K in the m–fold cyclic branched cover over K. We exhibit HFK̂(Σm(K);K˜,i) as the categorification of a well-defined multiple of the Turaev torsion of Σm(K) K˜ in the case where Σm(K) is a rational homology sphere. In addition, when K is a two-bridge knot, we prove that HFK̂(Σ2(K);K˜,s0)HFK̂(S3;K) for s0 the spin Spinc structure on Σ2(K). We conclude with a calculation involving two knots with identical HFK̂(S3;K,i) for which HFK̂(Σ2(K);K˜,i) differ as 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