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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Morse structures on open books
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by David T. Gay and Joan E. Licata PDF
Trans. Amer. Math. Soc. 370 (2018), 3771-3802 Request permission

Corrigendum: Trans. Amer. Math. Soc. 373 (2020), 3007-3008.


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
  • MR Author ID: 686652
  • Email:
  • Joan E. Licata
  • Affiliation: Mathematical Sciences Institute, The Australian National University, Canberra ACT0200, Australia
  • MR Author ID: 849641
  • Email:
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3771-3802
  • MSC (2010): Primary 53D10; Secondary 57M27, 57M50
  • DOI:
  • MathSciNet review: 3811509