## Algebraic & Geometric Topology

### Contact surgery and supporting open books

Russell Avdek

#### Abstract

Let $( M , ξ )$ be a contact 3–manifold. We present two new algorithms, the first of which converts an open book $( Σ , Φ )$ supporting $( M , ξ )$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $( M , ξ )$ into a supporting open book decomposition. These constructions lead to a refinement of a result of Ding and Geiges [Math. Proc. Cambridge Philos. Soc. 136 (2004) 583–598], which states that every such $( M , ξ )$ may be obtained by contact surgery from $( S 3 , ξ std )$, as well as bounds on the support norm and genus (Etnyre and Ozbagci [Trans. Amer. Math. Soc. 360 (2008) 3133–3151]) of contact manifolds obtained by surgery in terms of classical link data. We then introduce Kirby moves called ribbon moves, which use mapping class relations to modify contact surgery diagrams. Any two surgery diagrams of the same contact 3–manifold are related by a sequence of Legendrian isotopies and ribbon moves. As most of our results are computational in nature, a number of examples are analyzed.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 3 (2013), 1613-1660.

Dates
Accepted: 18 January 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715594

Digital Object Identifier
doi:10.2140/agt.2013.13.1613

Mathematical Reviews number (MathSciNet)
MR3071137

Zentralblatt MATH identifier
1275.57035

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

#### Citation

Avdek, Russell. Contact surgery and supporting open books. Algebr. Geom. Topol. 13 (2013), no. 3, 1613--1660. doi:10.2140/agt.2013.13.1613. https://projecteuclid.org/euclid.agt/1513715594

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