Non-braid-positive hyperbolic $L$-space knots

Keisuke Himeno

Vol. 342 (2026), No. 2, 275–297

DOI: 10.2140/pjm.2026.342.275

Abstract

An $L$-space knot is a knot that admits a positive Dehn surgery yielding an $L$-space. Many known hyperbolic $L$-space knots are braid positive, meaning they can be represented as the closure of a positive braid. Recently, Baker and Kegel showed that the hyperbolic $L$-space knot $o9\text{ \_}30634$ from Dunfield’s census is not braid-positive, and they constructed infinitely many candidates for hyperbolic $L$-space knots that may not be braid-positive. However, it remains unproven whether their examples are genuinely non-braid-positive. In this paper, we construct infinitely many hyperbolic $L$-space knots that are not braid-positive, and are distinct from those considered by Baker and Kegel.

Keywords

$L$-space knot, braid-positive

Mathematical Subject Classification

Primary: 57K10, 57K18

Milestones

Received: 7 July 2025

Revised: 5 November 2025

Accepted: 27 December 2025

Published: 23 April 2026