Hyperbolic L-space knots and their formal semigroups

Teragaito, Masakazu

doi:10.1142/s0129167x2250080x

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Two Infinite Families Of Hyperbolic L-space Knots Whose Alex...1

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2024

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“…Remark 4.3 The semigroups from Theorem 4.1 and the preceding remark all have rank 3, ie the minimal number of a generating set is 3. On the other hand, Teragaito constructs in [28] an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups of rank 5.…”

Section: Two Infinite Families Of Hyperbolic L-space Knots Whose Alex...mentioning

confidence: 99%

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

Baker,

Kegel

2024

Algebr. Geom. Topol.

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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.

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Section: Two Infinite Families Of Hyperbolic L-space Knots Whose Alex...mentioning

confidence: 99%