Tangent circle bundles admit positive open book decompositions along arbitrary links

Ishikawa, Masaharu

doi:10.1016/s0040-9383(03)00040-5

Cited by 12 publications

(33 citation statements)

References 20 publications

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“…This gives a contradiction since ξ can is the unique strongly symplectically fillable contact structure on T 3 by Eliashberg [4], and it is non-planar [5]. Here is another way to see the error in Ishikawa'a paper [10]. For Σ = S 2 , Ishikawa's method would give a Lefschetz fibration on the disk tangent bundle of S 2 , which is the D 2 -bundle over S 2 with Euler number +2.…”

Section: 2mentioning

confidence: 99%

Genus one Lefschetz fibrations on disk cotangent bundles of surfaces

Özbağcı

2020

Rev. Mat. Iberoam.

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Section: 2mentioning

confidence: 99%

Genus one Lefschetz fibrations on disk cotangent bundles of surfaces

Özbağcı

2020

Rev. Mat. Iberoam.

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“…This theorem was popularized by Fried [17] who named them Birkhoff sections. The theorem holds in even more generality as shown in [1,2,22,24]. As is clear from this rich history there are a number of proofs of this theorem especially for compact hyperbolic surfaces (see e.g.…”

Section: Linking Numbers In γ\Sl 2 (R)mentioning

confidence: 90%

“…(1 − q m ) 24 where as usual q = e(z) = e 2πiz for z ∈ H. Thus for any γ = a b c d ∈ Γ we have (1.6) log ∆(γz) − log ∆(z) = 6 log(−(cz + d)…”

Section: Introductionmentioning

confidence: 99%

Modular cocycles and linking numbers

Duke¹,

Imamoḡlu²,

Tóth³

2017

Duke Math. J.

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Abstract. It is known that the 3-manifold SL(2, Z)\ SL(2, R) is diffeomorphic to the complement of the trefoil knot in S 3 . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind's eta function under SL(2, Z). In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.

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“…The unit tangent bundle determines a canonical contact structure in φ −1 (D ) and we can think it as a part of the standard contact structure in S(T (R 2 )) = S 3 . See [9] for a precise explanation.…”

Section: Proofsmentioning

confidence: 99%

On the Thurston–Bennequin invariant of graph divide links

Ishikawa

2005

Math. Proc. Camb. Phil. Soc.

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We determine the Thurston-Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A. I. Stipsicz, we prove that the 3-manifold obtained from S 3 by Dehn surgery along a non-trivial graph divide knot K with coefficient r carries positive, tight contact structures for every r except the Thurston-Bennequin invariant of K.

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Tangent circle bundles admit positive open book decompositions along arbitrary links

Cited by 12 publications

References 20 publications

Genus one Lefschetz fibrations on disk cotangent bundles of surfaces

Genus one Lefschetz fibrations on disk cotangent bundles of surfaces

Modular cocycles and linking numbers

On the Thurston–Bennequin invariant of graph divide links

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