UCLA

Heegaard Floer homology is combinatorially computable, but a convenient computational scheme in general is still missing, especially for $HF^-$ of hyperbolic manifolds. We aim to use the Manolescu-Ozsv\'{a}th the link surgery formula for computing Heegaard Floer homology of surgeries on links and finding applications on $L$-space surgeries on links. The main difficulty is to reduce the complexity of the algorithms.

We give a polynomial time algorithm to compute the whole package of the completed

Heegaard Floer homology ${\textbf{HF}}^{-}$ of all surgeries on a two-bridge link of slope $q/p$, $L=b(p,q)$, by using nice diagrams and some algebraic rigidity results to simplify the link surgery formula.

We also initiate a general study of the definitions, properties, and examples of $L$-space links. In particular, we find many hyperbolic $L$-space links, including some chain links and two-bridge links; from them, we obtain many hyperbolic $L$-spaces by integral surgeries, including the Weeks manifold. We give bounds on the ranks of the link Floer homology of $L$-space links and on the coefficients in the multi-variable Alexander polynomials. We also describe the Floer homology of surgeries on any $L$-space link using the link surgery formula of Manolescu and Ozsv\'{a}th.

As applications, we compute the graded Heegaard Floer homology of surgeries on 2-component $L$-space links in terms of only the Alexander polynomial and the surgery framing. We also give a fast algorithm to classify $L$-space surgeries among them.