Essential open book foliations and fractional Dehn twist coefficient

Ito, Tetsuya; Kawamuro, Keiko

doi:10.1007/s10711-016-0188-7

Cited by 33 publications

(56 citation statements)

References 41 publications

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“…Very briefly, to define the fractional Dehn twist coefficient in this alternate way, one can consider the compactification of the universal cover of

D_{n}

embedded in

H^{2}

, use the action of the lift of

β

to this universal cover to define a map

Θ : B_{n} \to \tilde{H o m e o^{+} false(S^{1} false)}

, and define

ω (β)

to be the translation number of

Θ (β)

. For a more thorough discussion, see . For yet another alternate and equivalent definition that demonstrates more clearly that the fractional Dehn twist coefficient is measuring the amount of (signed) twisting a braid realizes around

\partial D_{n}

, see .…”

Section: Background On the Fractional Dehn Twist Coefficientmentioning

confidence: 99%

“…For a more thorough discussion, see . For yet another alternate and equivalent definition that demonstrates more clearly that the fractional Dehn twist coefficient is measuring the amount of (signed) twisting a braid realizes around

\partial D_{n}

, see . Both of these alternate definitions generalize easily beyond braids to elements in mapping class groups of surfaces with boundary.…”

Section: Background On the Fractional Dehn Twist Coefficientmentioning

confidence: 99%

“…The fractional Dehn twist coefficient has been extensively studied in the context of contact topology and open book decompositions (see, for instance, [4,[22][23][24][25]30]) and of course in the context of classical braids [35,36]. Relationships have also been explored between the fractional Dehn twist coefficient and monoids in the mapping class group [13,26], classical knot theory [30], and homological invariants of knots and 3-manifolds [22,42].…”

Section: Background On the Fractional Dehn Twist Coefficientmentioning

confidence: 99%

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Braids with as many full twists as strands realize the braid index

Feller

Hubbard

2019

Journal of Topology

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We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function‐valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that n‐braids with fractional Dehn twist coefficient larger than n−1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that n‐times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.

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“…Very briefly, to define the fractional Dehn twist coefficient in this alternate way, one can consider the compactification of the universal cover of

D_{n}

embedded in

H^{2}

, use the action of the lift of

β

to this universal cover to define a map

Θ : B_{n} \to \tilde{H o m e o^{+} false(S^{1} false)}

, and define

ω (β)

to be the translation number of

Θ (β)

\partial D_{n}

, see .…”

Section: Background On the Fractional Dehn Twist Coefficientmentioning

confidence: 99%

\partial D_{n}

, see . Both of these alternate definitions generalize easily beyond braids to elements in mapping class groups of surfaces with boundary.…”

Section: Background On the Fractional Dehn Twist Coefficientmentioning

confidence: 99%

Section: Background On the Fractional Dehn Twist Coefficientmentioning

confidence: 99%

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Braids with as many full twists as strands realize the braid index

Feller

Hubbard

2019

Journal of Topology

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“…Among other things, foliation change and exchange move introduced in [3,4], and various observations and techniques developed in proving Markov Theorem Without Stabilization (MTWS) [6,7] and usual Markov theorem [5] play crucial roles. In our proof, we use an open book foliation machinery developed in [11,13,14,15] which is a generalization of the braid foliation.…”

Section: Example 12 Let (Amentioning

confidence: 99%

On a relation between the self-linking number and the braid index of closed braids in open books

Ito¹

2018

Kyoto J. Math.

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“…As shown in [4,Corollary 4.17] the FTDC map c(−, C) : Mod(S) → Q is not a homomorphism but a quasi-morphism if the surface S has negative Euler characteristic. In order to prove Theorem 1.1 we first study general quasi-morphisms and obtain a monoid criterion (Theorem 2.2).…”

Section: Basic Study Of Quasi-morphismsmentioning

confidence: 99%