Non-braid positive hyperbolic $L$-space knots

TL;DR

This paper constructs infinitely many hyperbolic $L$-space knots that are not braid positive, expanding the known examples beyond previous constructions and addressing open questions about their properties.

Contribution

It provides new infinite families of hyperbolic $L$-space knots that are proven to be non-braid positive, distinct from prior examples.

Findings

Constructed infinitely many non-braid positive hyperbolic $L$-space knots.

Examples are proven to be distinct from previous known examples.

Addresses open questions about the braid positivity of certain hyperbolic $L$-space knots.

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 $o9\_30634$ 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 our examples are distinct from those considered by Baker and Kegel.