Twist families of $L$-space knots, their genera, and Seifert surgeries

Baker, Kenneth L.; Motegi, Kimihiko

doi:10.4310/cag.2019.v27.n4.a1

Cited by 5 publications

(18 citation statements)

References 44 publications

(104 reference statements)

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“…Let {K n } be a twist family of knots obtained by twisting K along c. If c links K coherently at least twice, then both (i) there exists constants C + , C − 0 such that g(K n ) − g 4 (K n ) = C ± for sufficiently large integers ±n, and (ii) g(K n )/g 4 (K n ) → 1 as |n| → ∞. Since ω = η > 1 in this theorem, g(K n ) → ∞ as |n| → ∞ by [5,Theorem 2.1]. Hence (i) implies (ii).…”

Section: Comparison Of Seifert Genera and Slice Genera Of Knots Under...mentioning

confidence: 89%

“…However, assume that

2 ⩽ q < p < 2 q

, and take

c_{-}

instead of

c_{+}

. Then by [, Proposition 6.3 ]

c_{-}

is not a braid axis of

T_{p, q}

, and the knot

T_{p, q, n}^{-}

obtained from

T_{p, q}

n

‐twist along

c_{-}

is an L‐space knot for

n ⩾ - 1

. Now Corollary allows us to conclude that

T_{p, q, n}^{-}

is an L‐space knot for only finitely many

n ⩽ - 2

.…”

Section: L‐space Knots In Twist Familiesmentioning

confidence: 99%

See 1 more Smart Citation

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

Baker

Motegi

2019

Proceedings of London Math Soc

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Twisting a knot K in S 3 along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. We prove that if the ratio of the Seifert genus to the slice genus for knots in a twist family limits to 1, then the winding number of K about c equals either zero or the wrapping number. As a key application, if {Kn} or the mirror twist family {Kn} contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that c is a braid axis of K if and only if both {Kn} and {Kn} each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for {Kn} to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore. Contents

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Section: Comparison Of Seifert Genera and Slice Genera Of Knots Under...mentioning

confidence: 89%

“…However, assume that

2 ⩽ q < p < 2 q

, and take

c_{-}

instead of

c_{+}

. Then by [, Proposition 6.3 ]

c_{-}

is not a braid axis of

T_{p, q}

, and the knot

T_{p, q, n}^{-}

obtained from

T_{p, q}

n

‐twist along

c_{-}

is an L‐space knot for

n ⩾ - 1

. Now Corollary allows us to conclude that

T_{p, q, n}^{-}

is an L‐space knot for only finitely many

n ⩽ - 2

.…”

Section: L‐space Knots In Twist Familiesmentioning

confidence: 99%

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

Baker

Motegi

2019

Proceedings of London Math Soc

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“…Hence n = 0. Note that the torus knot space E(K q,n ) = E(K q,n,0 ) = E(T 2,2q+1 ) is obtained from the solid torus V = S 3 − intN (K q,n,p ) by 1 p -surgery on c n . Since E(K q,n ) is a Seifert fiber space and V −intN (c n ) is hyperbolic (Claim 3.1), we may apply [13, Theorem 1.2] to conclude that |p| = 1.…”

Section: Generalized Torsion Which Arises From Twisting Torus Knots T...mentioning

confidence: 99%

Generalized torsion for knots with arbitrarily high genus

Motegi¹,

Teragaito²

2021

Preprint

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Let G be a group and let g be a non-trivial element in G. If some non-empty finite product of conjugates of g equals to the identity, then g is called a generalized torsion element. We say that a knot K has generalized torsion if G(K) = π 1 (S 3 − K) admits such an element. For a (2, 2q + 1)-torus knot K, we demonstrate that there are infinitely many unknots cn in S 3 such that p-twisting K about cn yields a twist family {Kq,n,p} p∈Z in which Kq,n,p is a hyperbolic knot with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the (−2, 3, 7)-pretzel knot, have generalized torsion. Since generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

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“…This implies that 𝐾 𝑞,𝑛,±1 is nontrivial. Remark 3.7 (1) When 𝑞 = 0, the link 𝐾 0 ∪ 𝑐 𝑛 is equivalent to the pretzel link of type (−2, 3, 2𝑛 + 6). This special case is treated in [20].…”

Section: Remark 34mentioning

confidence: 99%

“…Then 𝐾 𝑞,𝑛, 𝑝 is a hyperbolic knot with a generalized torsion element whenever | 𝑝| > 3. Since the linking number between 𝐾 𝑞 and 𝑐 𝑛 is greater than one,[1, Theorem 2.1] shows that the genus of 𝐾 𝑞,𝑛, 𝑝 tends to ∞ as 𝑝 → ∞. ■…”

mentioning

confidence: 99%

Generalized torsion for knots with arbitrarily high genus

Motegi¹,

Teragaito

2021

Can. Math. Bull.

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Let 𝐺 be a group and let 𝑔 be a non-trivial element in 𝐺. If some non-empty finite product of conjugates of 𝑔 equals to the identity, then 𝑔 is called a generalized torsion element. We say that a knot 𝐾 has generalized torsion if 𝐺 (𝐾 ) = 𝜋 1 (𝑆 3 − 𝐾 ) admits such an element. For a (2, 2𝑞 + 1)-torus knot 𝐾, we demonstrate that there are infinitely many unknots 𝑐 𝑛 in 𝑆 3 such that 𝑝-twisting 𝐾 about 𝑐 𝑛 yields a twist family {𝐾 𝑞,𝑛, 𝑝 } 𝑝∈Z in which 𝐾 𝑞,𝑛, 𝑝 is a hyperbolic knot with generalized torsion whenever | 𝑝 | > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the (−2, 3, 7)-pretzel knot, have generalized torsion. Since generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

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Twist families of $L$-space knots, their genera, and Seifert surgeries

Cited by 5 publications

References 44 publications

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

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

Generalized torsion for knots with arbitrarily high genus

Generalized torsion for knots with arbitrarily high genus

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