# L-space surgeries on 2-component L-space links

## TL;DR

This paper characterizes 2-component L-space links based on their L-space surgeries, identifying conditions under which the link is an unknot, Hopf link, or torus link, and provides explicit surgery set descriptions.

## Contribution

It introduces new characterizations of 2-component L-space links using L-space surgeries, including conditions for unknot, Hopf, and torus links, with explicit surgery set descriptions.

## Key findings

- Negative surgery coefficient implies unknotted component.
- Very negative L-space surgeries identify the Hopf link.
- Certain L-space surgeries characterize torus links T(2, 2l).

## Abstract

In this paper, we analyze L-space surgeries on two component L-space links. We show that if one surgery coefficient is negative for the L-space surgery, then the corresponding link component is an unknot. If the link admits very negative (i.e. $d_{1}, d_{2}\ll0$) L-space surgeries, it is the Hopf link. We also give a way to characterize the torus link $T(2, 2l)$ by observing an L-space surgery $S^{3}_{d_{1}, d_{2}}(\mathcal{L})$ with $d_{1}d_{2}<0$ on a 2-component L-space link with unknotted components. For some 2-component L-space links, we give explicit descriptions of the L-space surgery sets.