Applications of the Link Surgery Formula in Heegaard Floer Homology

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-Ozsvath 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 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 Ozsvath.;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.