$L$-space surgeries on links

TL;DR

This paper introduces the concept of L-space links, explores their properties and examples, and provides methods to compute their Floer homology and classify L-space surgeries, expanding understanding of hyperbolic 3-manifolds.

Contribution

It initiates a comprehensive study of L-space links, including their properties, examples, and computational techniques for Floer homology and surgery classification.

Findings

Many hyperbolic L-space links identified, including chain and two-bridge links.

Bounds established on link Floer homology ranks and Alexander polynomial coefficients.

Provided algorithms for classifying L-space surgeries and computing Floer homology.

Abstract

An $L$-space link is a link in $S^3$ on which all large surgeries are $L$-spaces. In this paper, we 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 Ozsv\'{a}th and Manolescu. 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, and give a fast algorithm to classify $L$-space surgeries among them.