Morse structures on open books
David T Gay, Joan E. Licata · Transactions of the American Mathematical Society · 2016
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.