Morse structures on open books

Gay, David; Licata, Joan

doi:10.1090/tran/7079

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 -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.

[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, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. MR 3012475
[3] John B. Etnyre, Legendrian and transversal knots, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 105–185. MR 2179261, https://doi.org/10.1016/B978-044451452-3/50004-6
[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, https://doi.org/10.1090/S0002-9947-08-04459-0
[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, https://doi.org/10.2140/gt.2015.19.2465
[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, https://doi.org/10.2140/pjm.2011.253.475
[11] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR 0375366, https://doi.org/10.1090/S0002-9939-1975-0375366-7
[12] Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, https://doi.org/10.14492/hokmj/1381413841