Morse Structures on Open Book Decompositions with Punctured Torus Pages
Creators: name: Brand, Jack
Publisher: name: The Australian National University
Year: 2023

description: In this thesis, we use Morse structures to study contact 3-manifolds, and Legendrian knots embedded within. In particular, we focus on contact 3-manifolds whose compatible open book decompositions have punctured torus pages. Morse structures on an open book decomposition were introduced by Licata and Gay in 'Morse structures on open books' and are a powerful tool for studying contact 3-manifolds. From a Morse structure we can construct a Morse diagram that encodes the contact manifold. Moreover, we can study Legendrian knots via a front projection onto these diagrams. Amongst other things, Morse structures have been used to compute the homology, Thurston-Bennequin number, and rotation number of Legendrian knots, and can be used to combinatorially compute the homology of the underlying 3-manifold, and the Euler class of the contact structure of the contact manifold. We continue to extract information from Morse structures. In the first part of this thesis, we show how to take a Murasugi sum of two contact open books with compatible Morse structures. Using this construction, we then show how a Morse diagram can be used to give a criterion for overtwistedness of a contact 3-manifold. Next, in the case of open books with puncture torus pages, we show that standard factorisations of the monodromy of the open book are closely related to the Morse diagram. Using the constructions of this part, we are able to construct many new Morse diagrams by taking Morse diagrams from already known examples, and by combining these by a Murasugi sum operation. Additionally, for open books with punctured torus pages, by comparing Morse diagrams to the typical presentation of the mapping class group of the torus with boundary, we can construct even more examples. In the second part of this thesis, in the case of an open book with punctured torus pages, we construct two notions of a Lagrangian diagram of a Legendrian knot. The first is a more complicated diagram, but can easily be translated from a front projection of the knot. The second diagram is simpler, but requires us to manipulate our Legendrian knot so that it is in a nice position with respect to a page of the open book. This is motivated by the third part of this thesis, where we use this diagram to define a Legendrian knot invariant (a differential graded algebra) motivated by symplectic field theory. At the time of writing, a Chekanov-Eliashberg type of DGA has only been defined for a handful of strict contact 3-manifolds. The definition of such a DGA relies heavily on the understanding of the Reeb dynamics of the contact manifold, which are notoriously difficult to understand. By using a Morse structure, we can define a contactomorphism from `most' of the contact manifold to a contact manifold with simpler Reeb dynamics. For contact 3-manifolds with compatible open book with punctured torus pages, we define a DGA in the image of this contactomorphism. This invariant has a multiple benefits. Firstly, it is applicable to a wide class if Legendrian knots, including knots that are not nullhomologous, and secondly it is applicable in a wide class of contact 3-manifolds. We provide examples of computing this invariant in the 3-sphere, but not with the standard contact tight contact structure, and in contact 3-manifolds that are not Seifert fibered.