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.
Keywords
Cite
@article{arxiv.2203.12013,
title = {Census L-space knots are braid positive, except for one that is not},
author = {Kenneth L. Baker and Marc Kegel},
journal= {arXiv preprint arXiv:2203.12013},
year = {2026}
}
Comments
The main article is just 12 pages. The remaining 31 pages is a listing of braid words for the L-space knots in the SnapPy census. Except for o9_30634, these braid words are either positive or negative according to the orientation of the manifold in the census. The auxiliary files include 4 supporting jupyter notebooks of calculations. V2 adds a new section of applications