Census L-space knots are braid positive, except for one that is not

TL;DR

This paper demonstrates that all L-space knots in the SnapPy census are braid positive except one, introduces a family of hyperbolic L-space knots with unique properties, and disproves a conjecture regarding Alexander polynomial roots.

Contribution

It provides the first examples of hyperbolic L-space knots with actual semigroups and roots of Alexander polynomials at roots of unity, challenging previous conjectures.

Findings

All but one L-space knot in the census are braid positive.

A family of hyperbolic L-space knots with actual semigroups is constructed.

The roots of Alexander polynomials are roots of unity, disproving a conjecture.

Abstract

We exhibit braid positive presentations for all L-space knots in the SnapPy census except one, which is not braid positive. The normalized HOMFLY polynomial of o9_30634, when suitably normalized is not positive, failing a condition of Ito for braid positive knots. We generalize this knot to a 1-parameter family of hyperbolic L-space knots that might not be braid positive. Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic L-space knots whose formal semigroups are actual semigroups, answering a question of Wang. Furthermore, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li-Ni.