Algebraic & Geometric Topology

Floer homology and splicing knot complements

Eaman Eftekhary

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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.

Article information

Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3155-3213.

Received: 4 November 2013
Revised: 19 February 2015
Accepted: 1 March 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

Floer homology splicing essential torus


Eftekhary, Eaman. Floer homology and splicing knot complements. Algebr. Geom. Topol. 15 (2015), no. 6, 3155--3213. doi:10.2140/agt.2015.15.3155.

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