Heegaard Floer homology of surgeries on two-bridge links
\textit{C. Manolescu} and \textit{P. Ozsváth} [``Heegaard Floer homology and integer surgeries on links'', Preprint, \url{arXiv:1011.1317}] developed a formula for the Heegaard Floer homology of surgeries on links. The author uses this formula to give a practical polynomial time algorithm to compute the Floer homology of surgeries on 2-bridge links. The algorithm mainly utilizes genus-0 nice diagrams and algebraic arguments to simplify the Manolescu-Ozsváth formula. In fact, two sets of data are needed in the surgery formula: the generalized Floer complex \(A^-_s(\vec{L})\) for a link \(L\), and the maps \(\mathcal{I}^{\vec{L'}}_s\) and \(D^{\vec{L'}}_s\) connecting the complexes associated to oriented sublinks \(L' \subset L\). Here is what the author does to simplify the Manolescu-Ozsváth formula: 1. The Schubert normal form of 2-bridge links is utilized to get a nice Heegaard diagram. With the nice diagram, the Floer complex \(A^-_s(\vec{L})\) can be directly computed. 2. The author defines a twisted gluing of squares of chain complexes. Then the link surgery formula is a twisted gluing of certain squares of chain complexes derived from the link \(L\). 3. The author shows some rigidity results of maps \(D^{\vec{L'}}_s\) up to chain homotopy for 2-bridge links, which mean perturbing the maps \(D^{\pm L'}_s\) by chain homotopy does not change the isomorphism type of the original complex. The author also shows that \(\boldsymbol{HF}^-\) and \(d\)-invariants of surgeries on a 2-bridge link \(L\) are determined only by the homotopy type of \(A^-_s(\vec{L})\). Some examples are computed in this paper, which provide new examples of hyperbolic 3-manifolds whose Heegaard Floer homology can be computed.
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