Heegaard Floer homology and several families of Brieskorn spheres

TL;DR

This paper extends the computation of Heegaard Floer homology invariants to new families of Brieskorn spheres, generalizing previous formulas and providing explicit calculations for specific cases.

Contribution

It introduces a new formula for HF^{+} of -{ extSigma}(p,q,pqn-1) and computes HF^{+} for the families -{ extSigma}(2,5,k) and -{ extSigma}(2,7,k), expanding the understanding of these invariants.

Findings

Derived a formula for HF^{+}(-{ extSigma}(p,q,pqn-1)).

Computed HF^{+} for -{ extSigma}(2,5,k).

Computed HF^{+} for -{ extSigma}(2,7,k).

Abstract

Ozsv\'ath and Szab\'o gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. N\'emethi constructed a remarkable algorithm for executing these computations for almost-rational plumbings, and his work gives a formula computing the invariants for the Brieskorn homology spheres -{\Sigma} (p,q,pqn + 1). Here we give a formula for HF^{+}(-{\Sigma}(p,q,pqn-1)), generalizing the one for the n=1 case given by Borodzik and N\'emethi. We also compute HF^{+} for the families -{\Sigma}(2,5,k) and -{\Sigma}(2,7,k).