Floer homology and splicing knot complements

Abstract

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y\left(K_1,K_2\right)$ 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 $\widehat{HFK}$ for the knot obtained by $n$–surgery over $K_i$, we show that the rank of $\widehat{HF}\left(Y\left(K_1,K_2\right)\right)$ is bounded below by

$$|\left(h_{\infty }^1-h_1^1\right)\left(h_{\infty }^2-h_1^2\right)-\left(h_0^1-h_1^1\right)\left(h_0^2-h_1^2\right)|.$$

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.