Morse structures on open books

TL;DR

This paper introduces a Morse-theoretic approach to encode contact topology on open books, generalizing Legendrian front projections and providing combinatorial tools for knot invariants in contact 3-manifolds.

Contribution

It develops a Morse-theoretic framework to represent contact structures on open books and extends Legendrian front projections beyond standard settings.

Findings

Morse-theoretic encoding of contact topology on open books

Extension of Legendrian front projections to arbitrary contact 3-manifolds

A combinatorial formula for Thurston-Bennequin number from front diagrams

Abstract

We use parameterized Morse theory on the pages of an open book decomposition 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.