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 $K_isubset Y_i$, $i=1,2$, 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 $widehat{mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $widehat{mathrm{HF}}(Y(K_1,K_2))$ 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 $Ksubset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.
We show that if a prime homology sphere has the same Floer homology as the standard three-sphere, it does not contain any incompressible tori.
We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y.
Given a symplectic three-fold $(M,omega)$ we show that for a generic almost complex structure $J$ which is compatible with $omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $ggeq 0$ representing a homology class $beta$ in $H
_2(M,Z)$ with $c_1(M).beta=0$, provided that the divisibility of $beta$ is at most 4 (i.e. if $beta=nalpha$ with $alphain H_2(M,Z)$ and $nin Z$ then $nleq 4$). Moreover, each such curve is embedded and 4-rigid.
This paper has been withdrawn by the author, due a crucial mistake in proof of lemma 4.2.
Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.
We give a precise description of splicing formulas from a previous paper in terms of knot Floer complex associated with a knot in homology sphere.
