An Excision Theorem in Heegaard Floer Theory

Let $Y_1$ be a closed, oriented 3-manifold and $\Sigma$ denote a non-separating closed, orientable surface in $Y_1$ which consists of two connected components of the same genus. By cutting $Y_1$ along $\Sigma$ and re-gluing it using an orientation-preserving diffeomorphism of $\Sigma$ we obtain another closed, oriented 3-manifold $Y_2$. When the excision surface $\Sigma$ is of genus one, we show that twisted Heegaard Floer homology groups of $Y_1$ and $Y_2$ (twisted with coefficients in the universal Novikov ring) are isomorphic. We use this excision theorem to demonstrate that certain manifolds are not related by the excision construction on a genus one surface. Additionally, we apply the excision formula to compute twisted Heegaard Floer homology groups of 0-surgery on certain two-component links, including some families of 2-bridge links.