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

Baker, Kenneth L; Kegel, Marc

doi:10.2140/agt.2024.24.569

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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

DOI: 10.2140/agt.2024.24.569

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 $o 9_30634$ , 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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Volume 24, issue 1 (2024)