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

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.

The main article is just 12 pages. The remaining pages are a listing of braid words for the L-space knots in the SnapPy census. The auxiliary files include 4 supporting jupyter notebooks of calculations. V2 adds a new section of applications; v3: Updated to the published version; added a note that our examples give a negative solution to Problem 1.21(c)(iii) from the K3 Kirby list

References in corpus (4)

• Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds
• A note on HOMFLY polynomial of positive braid links
• L-space knots with tunnel number >1 by experiment
• Computing a Link Diagram from its Exterior

Cited by in corpus (3)

• Twist positivity, L-space knots, and concordance
• Two curious strongly invertible L-space knots
• Hyperbolic L-space knots not concordant to algebraic knots