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We obtain a formula for the Heegaard Floer homology (hat theory) of
the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the
knots $K_i\subset Y_i$, $i=1,2$, in terms of the knot Floer
homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes
the rank of the Heegaard Floer group $\ov\HFKT$ for the knot obtained by $n$-surgery
over $K_i$ we show that the rank of $\ov{\HFT}(Y(K_1,K_2))$ is bounded below by
$$\big|(h_\infty^1-h_1^1)(h_\infty^2-h_1^2)-
(h_0^1-h_1^1)(h_0^2-h_1^2)\big|.$$
We also show that if splicing the complement of a knot $K\subset Y$ 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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