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

Kenneth L Baker and Marc Kegel

Algebraic & Geometric Topology 24 (2024) 569–586
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 may 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. Further, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li and Ni.

Keywords
L–space knots, braid positivity, SnapPy census knots, formal semigroups of L–space knots
Mathematical Subject Classification
Primary: 57K10
Secondary: 57M12, 57R65
Supplementary material

Braid words for all SnapPy census knots

References
Publication
Received: 17 May 2022
Revised: 1 July 2022
Accepted: 6 September 2022
Published: 18 March 2024
Authors
Kenneth L Baker
Department of Mathematics
University of Miami
Coral Gables, FL
United States
Marc Kegel
Mathematisches Institut
Humboldt-Universität zu Berlin
Berlin
Germany

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