Geodesic
Census L–space knots are braid positive, except for one that is not
Baker, Kenneth L1 ; Kegel, Marc2
1 Department of Mathematics, University of Miami, Coral Gables, FL, United States
2 Mathematisches Institut, Humboldt-Universität zu Berlin, Berlin, Germany
Algebraic and Geometric Topology, Tome 24 (2024) no. 1, pp. 569-586

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/agt.2024.24.569
Keywords: L–space knots, braid positivity, SnapPy census knots, formal semigroups of L–space knots

Baker, Kenneth L 1 ; Kegel, Marc 2

1 Department of Mathematics, University of Miami, Coral Gables, FL, United States
2 Mathematisches Institut, Humboldt-Universität zu Berlin, Berlin, Germany 
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Baker, Kenneth L; Kegel, Marc. Census L–space knots are braid positive, except for one that is not. Algebraic and Geometric Topology, Tome 24 (2024) no. 1, pp. 569-586. doi: 10.2140/agt.2024.24.569

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