#### Volume 18, issue 3 (2014)

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Open book foliation

### Tetsuya Ito and Keiko Kawamuro

Geometry & Topology 18 (2014) 1581–1634
##### Abstract

We study open book foliations on surfaces in $3$–manifolds and give applications to contact geometry of dimension $3$. We prove a braid-theoretic formula for the self-linking number of transverse links, which reveals an unexpected connection with to the Johnson–Morita homomorphism in mapping class group theory. We also give an alternative combinatorial proof of the Bennequin–Eliashberg inequality.

##### Keywords
open book decomposition, contact structure, self-linking number, Johnson–Morita homomorphism
##### Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M50, 57R17, 53D35
##### Publication
Received: 25 January 2013
Revised: 3 September 2013
Accepted: 19 October 2013
Published: 7 July 2014
Proposed: Shigeyuki Morita
Seconded: Yasha Eliashberg, Walter Neumann
##### Authors
 Tetsuya Ito Research Institute for Mathematical Sciences Kyoto University Sakyo-ku Kyoto 606-8502 Japan http://kurims.kyoto-u.ac.jp/~tetitoh/ Keiko Kawamuro Department of Mathematics The University of Iowa 14 McLean Hall Iowa City, IA 52242 USA