Volume 13, issue 3 (2013)

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Contact surgery and supporting open books

Russell Avdek

Algebraic & Geometric Topology 13 (2013) 1613–1660

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 (S3,ξstd), 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.

contact structure, contact surgery, open book
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 57M25
Received: 22 October 2012
Accepted: 18 January 2013
Published: 16 May 2013
Russell Avdek
Department of Mathematics
University of Southern California
KAP 406a
3620 S. Vermont Ave.
Los Angeles, CA 90089