Contact surgery and supporting open books

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.

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Keywords

contact structure, contact surgery, open book

Mathematical Subject Classification 2010

Primary: 57R17

Secondary: 57M25

References

Publication

Received: 22 October 2012

Accepted: 18 January 2013

Published: 16 May 2013

Authors

Russell Avdek
Department of Mathematics University of Southern California KAP 406a 3620 S. Vermont Ave. Los Angeles, CA 90089 USA
http://www-scf.usc.edu/~avdek/

Volume 13, issue 3 (2013)

Russell Avdek

Abstract