(1,1) L-space knots

TL;DR

This paper characterizes (1,1) knots in the 3-sphere and lens spaces that allow non-trivial L-space surgeries, revealing new insights into the structure of these knots and their relation to 1-bridge braids.

Contribution

It provides a complete characterization of (1,1) knots admitting non-trivial L-space surgeries and links this to 1-bridge braids and Berge manifolds.

Findings

(1,1) knots in S^3 and lens spaces admit non-trivial L-space surgeries

1-bridge braids in these manifolds also admit such surgeries

Berge manifold characterized among 1-bridge braid exteriors

Abstract

We characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non- trivial L-space surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors.