Open book decompositions of fiber sums in contact topology

Klukas, Mirko

doi:10.2140/agt.2016.16.1253

Cited by 2 publications

(6 citation statements)

References 12 publications

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“…We discuss both regular fibre sums and fibre sums of contact manifolds. While, at first glance, the binding sum destroys the open book structure, we show that this is in fact not the case and generalise a previous result of the second author [16] to higher dimensions. The main results can be summarized as follows:…”

Section: Introductionsupporting

confidence: 58%

“…It is also worth comparing this with the situation in one dimension lower as discussed in [16], i.e. the binding sum of two copies of (D 2 , id).…”

Section: Proof Of the Main Resultsmentioning

confidence: 93%

“…with page framing (here, a framing has to be specified). Then the new page is an annulus and the monodromy is isotopic to the identity: the Chinese burn part of the monodromy are two trivial negative Dehn twists and two non-trivial positive Dehn twists which cancel with the two positive Dehn twists forming the twist map (see [16,Theorem 3]).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…The new binding is the contact binding sum of the B i with respect to the specified open book decomposition. Hence, the new binding has an open book decomposition with page an annulus and -applying the formula from [16] for the page framing -monodromy isotopic to the identity. This means that the resulting binding is S 2 × S 1 with standard contact structure.…”

Section: Application To Contact Topologymentioning

confidence: 99%

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Nested open books and the binding sum

Durst,

Klukas

2016

Preprint

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We introduce the notion of a nested open book, a submanifold equipped with an open book structure compatible with an ambient open book, and describe in detail the special case of a push-off of the binding of an open book. This enables us to explicitly describe a natural open book decomposition of a fibre connected sum of two open books along their bindings, provided they are diffeomorphic and admit an open book structure themselves. Furthermore, we apply the results to contact open books, showing that the natural open book structure of a contact fibre connected sum of two adapted open books along their contactomorphic bindings is again adapted to the resulting contact structure.

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Section: Introductionsupporting

confidence: 58%

“…It is also worth comparing this with the situation in one dimension lower as discussed in [16], i.e. the binding sum of two copies of (D 2 , id).…”

Section: Proof Of the Main Resultsmentioning

confidence: 93%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Application To Contact Topologymentioning

confidence: 99%

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Nested open books and the binding sum

Durst,

Klukas

2016

Preprint

Self Cite

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“…pq`p´npqp1q`pnp´qqp1q pq`p´npqp0q`pnp´qqp´1q " p q´np . [36]. The construction of Baker, Etnyre, and van Horn-Morris in [2] does not cover the case r ą 0.…”

Section: 32mentioning

confidence: 99%

Transverse Surgery on Knots in Contact 3-Manifolds

Conway

2019

Trans. Amer. Math. Soc.

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We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. The overarching theme of this paper is to show that in many contexts, surgery on transverse knots is more natural than surgery on Legendrian knots.Besides reinterpreting surgery on Legendrian knots in terms of transverse knots, our main results on are in two complementary directions: conditions under which inadmissible transverse surgery (cf. positive contact surgery on Legendrian knots) preserves tightness, and conditions under which it creates overtwistedness. In the first direction, we give the first result on the tightness of inadmissible transverse surgery for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g´1. In the second direction, along with more general statements, we deduce a partial generalisation to a result of Lisca and Stipsicz: when L is a Legendrian knot with tbpLq ď´2, and |rotpLq| ě 2gpLq`tbpLq, then contact p`1q-surgery on L is overtwisted. arXiv:1409.7077v3 [math.GT]

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Open book decompositions of fiber sums in contact topology

Cited by 2 publications

References 12 publications

Nested open books and the binding sum

Nested open books and the binding sum

Transverse Surgery on Knots in Contact 3-Manifolds

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