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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Morse structures on open books


Authors: David T. Gay and Joan E. Licata
Journal: Trans. Amer. Math. Soc. 370 (2018), 3771-3802
MSC (2010): Primary 53D10; Secondary 57M27, 57M50
DOI: https://doi.org/10.1090/tran/7079
Published electronically: February 14, 2018
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Abstract: We use parameterized Morse theory on the pages of an open book decomposition supporting a contact structure to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standard contact $ \mathbb{R}^3$ to arbitrary closed contact $ 3$-manifolds. We describe a complete set of moves on such front diagrams, extending the standard Legendrian Reidemeister moves, and we give a combinatorial formula to compute the Thurston-Bennequin number of a nullhomologous Legendrian knot from its front projection.


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Additional Information

David T. Gay
Affiliation: Department of Mathematics, Euclid Lab, University of Georgia, Athens, Georgia 30606
Email: d.gay@euclidlab.org

Joan E. Licata
Affiliation: Mathematical Sciences Institute, The Australian National University, Canberra ACT0200, Australia
Email: joan.licata@anu.edu.au

DOI: https://doi.org/10.1090/tran/7079
Received by editor(s): September 4, 2015
Received by editor(s) in revised form: May 25, 2016, and August 1, 2016
Published electronically: February 14, 2018
Additional Notes: The first author was partially supported by NSF grant DMS-1207721.
Article copyright: © Copyright 2018 American Mathematical Society

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