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Floer homology and splicing knot complements

Eaman Eftekhary

Algebraic & Geometric Topology 15 (2015) 3155–3213
Abstract

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold Y (K1,K2) obtained by splicing the complements of the knots Ki Y i, i = 1,2, in terms of the knot Floer homology of K1 and K2. We also present a few applications. If hni denotes the rank of the Heegaard Floer group HFK̂ for the knot obtained by n–surgery over Ki, we show that the rank of HF̂(Y (K1,K2)) is bounded below by

|(h1 h 11)(h 2 h 12) (h 01 h 11)(h 02 h 12)|.

We also show that if splicing the complement of a knot K Y with the trefoil complements gives a homology sphere L–space, then K is trivial and Y is a homology sphere L–space.

Keywords
Floer homology, splicing, essential torus
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57R58
References
Publication
Received: 4 November 2013
Revised: 19 February 2015
Accepted: 1 March 2015
Published: 12 January 2016
Authors
Eaman Eftekhary
School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
PO Box 19395-5746
Tehran 19395
Iran
http://math.ipm.ir/~eftekhary