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IPM
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“School of Mathematics”

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Paper   IPM / M / 16148
School of Mathematics
  Title:   Floer homology and splicing knot complements
  Author(s):  Eaman Eftekhary
  Status:   Published
  Journal: Algebraic and Geometric Topology
  Vol.:  15
  Year:  2015
  Pages:   3155-3213
  Supported by:  IPM
  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 KiYi, 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 \ov\HFKT for the knot obtained by n-surgery over Ki we show that the rank of \ov\HFT(Y(K1,K2)) is bounded below by

(h1h11)(h2h12)−(h01h11)(h02h12)
.
We also show that if splicing the complement of a knot KY with the trefoil complements gives a homology sphere L-space then K is trivial and Y is a homology sphere L-space.

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