Seifert vs. slice genera of knots in twist families and a characterization of braid axes

Let L be an oriented link such that Σn(L), the n-fold cyclic cover of S 3 branched over L, is an L-space for some n 2. We show that if either L is a strongly quasi-positive link other than one with Alexander polynomial a multiple of (t − 1) 2g(L)+(|L|−1) , or L is a quasi-positive link other than one with Alexander polynomial divisible by (t − 1) 2g 4 (L)+(|L|−1) , then there is an integer n(L), determined by the Alexander polynomial of L in the first case and the Alexander polynomial of L and the smooth 4-genus of L, g4(L), in the second, such that n n(L). If K is a strongly quasi-positive knot with monic Alexander polynomial such as an L-space knot, we show that Σn(K) is not an L-space for n 6, and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if Σn(K) is an L-space for some n = 2, 3, 4, 5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating quasi-positive links. They also allow us to classify strongly quasi-positive alternating links and 3-strand pretzel links.

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“…Proof Baker and Motegi show that satellite L‐space knots can be expressed as a satellite knot where the pattern

P

is a braid [, Theorem 7.4]. In this case, the winding number of

P

is at least two in absolute value, so Proposition (a) implies that no

{normalΣ}_{n} false(K false)

is an L‐space.

□

…”

Section: Strongly Quasi‐positive Links With L‐space Branched Coversmentioning

confidence: 99%

Branched covers of quasi‐positive links and L‐spaces

Boileau

Boyer

Gordon

2019

Journal of Topology

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“…Moreover, the companion is a 2-bridge knot and the pattern knot has wrapping number two. We know that both of the companion and the pattern knot are L-space knots and the pattern is braided by [5,16]. Thus the companion is a 2-bridge torus knot [27], and K i is its 2-cable.…”

Section: Alexander Polynomialsmentioning

confidence: 99%

Hyperbolic L-space knots and their formal semigroups

Teragaito

2022

Int. J. Math.

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For an L-space knot, the formal semigroup is defined from its Alexander polynomial. It is not necessarily a semigroup. That is, it may not be closed under addition. There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by three elements. In this paper, we give the first infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by five elements.

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“…In this section, we calculate the Alexander polynomial ∆ Kn (t) of K n , and its formal semigroup. For the former, we mimic the argument in [2,5].…”

Section: Alexander Polynomials and Formal Semigroupsmentioning

confidence: 99%

“…Then [10] implies that the pattern knot is also an L-space knot. Furthermore, [5,Theorem 1.17] claims that the pattern is braided in the pattern solid torus. In particular, the wrapping number coincides with the winding number there.…”

Section: Hyperbolicitymentioning

confidence: 99%