# L-space surgeries on satellites by algebraic links

## TL;DR

This paper characterizes the space of rational surgeries yielding L-spaces for multi-component links, extending previous results to complex satellites and algebraic links, and develops tools for approximating these spaces.

## Contribution

It provides the first explicit descriptions of L-space surgery spaces for multi-component links and generalizes existing results to satellites by algebraic and iterated torus links.

## Key findings

- Computed L-space surgery spaces for satellites by torus-links in S^3.
- Developed approximation tools for fractal-boundaried L-space sets from algebraic links.
- Extended the L-space conjecture validity to satellite links of knots.

## Abstract

Given an $n$-component link $L$ in any 3-manifold $M$, the space $\mathcal{L} \subset (\mathbb{Q}\cup \mkern-1.5mu\{\infty\})^n$ of rational surgery slopes yielding L-spaces is already fully characterized (in joint work by the author) when $n\!=\!1$ and $\mathcal{L}$ is nontrivial. For $n\mkern-2mu>\mkern-3mu1$, however, there are no previous results for $\mathcal{L}$ as a rational subspace, and only limited results for integer surgeries $\mathcal{L}\cap\mathbb{Z}^n$ on $S^3\mkern-2mu$. Herein, we provide the first nontrivial explicit descriptions of $\mathcal{L}$ for rational surgeries on multi-component links. Generalizing Hedden's and Hom's L-space result for cables, we compute both $\mathcal{L}$, and its topology, for all satellites by torus-links in $S^3\mkern-2mu$. For fractal-boundaried $\mathcal{L}$ resulting from satellites by algebraic links or iterated torus links, we develop arbitrarily precise approximation tools. We also extend the provisional validity of the L-space conjecture for rational surgeries on a knot $K \subset S^3$ to rational surgeries on such satellite-links of $K$. These results exploit the author's generalized Jankins-Neumann formula for graph manifolds.