Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Published electronically: February 14, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Kenneth L. Baker and J. Elisenda Grigsby, Grid diagrams and Legendrian lens space links, J. Symplectic Geom. 7 (2009), no. 4, 415-448. MR 2552000
  • [2] Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and back: Symplectic geometry of affine complex manifolds, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. MR 3012475
  • [3] John B. Etnyre, Legendrian and transversal knots, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 105-185. MR 2179261,
  • [4] John B. Etnyre and Burak Ozbagci, Invariants of contact structures from open books, Trans. Amer. Math. Soc. 360 (2008), no. 6, 3133-3151. MR 2379791,
  • [5] Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
  • [6] David T. Gay and Robion Kirby, Indefinite Morse 2-functions: broken fibrations and generalizations, Geom. Topol. 19 (2015), no. 5, 2465-2534. MR 3416108,
  • [7] Emmanuel Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 405-414 (French, with French summary). MR 1957051
  • [8] W. B. R. Lickorish, A finite set of generators for the homeotopy group of a $ 2$-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769-778. MR 0171269
  • [9] Burak Ozbagci and András I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13, Springer-Verlag, Berlin; János Bolyai Mathematical Society, Budapest, 2004. MR 2114165
  • [10] Elena Pavelescu, Braiding knots in contact 3-manifolds, Pacific J. Math. 253 (2011), no. 2, 475-487. MR 2878820,
  • [11] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345-347. MR 0375366,
  • [12] Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241-251. MR 1114405,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53D10, 57M27, 57M50

Retrieve articles in all journals with MSC (2010): 53D10, 57M27, 57M50

Additional Information

David T. Gay
Affiliation: Department of Mathematics, Euclid Lab, University of Georgia, Athens, Georgia 30606

Joan E. Licata
Affiliation: Mathematical Sciences Institute, The Australian National University, Canberra ACT0200, Australia

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

American Mathematical Society