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 ⊂ Yi, 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
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(h∞1−h11)(h∞2−h12)−(h01−h11)(h02−h12) | ⎢
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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.
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