On September 25, 2017 I gave a talk as part of the Bill Tutte centenary celebration at Monash University.
This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.
Joint with Joan Licata.
We extend the notion of Morse structure on an open book to extendable partial open books in order to study contact 3-manifolds with convex boundary.
On 6 December, 2016, I gave a talk at the Austrlaian Mathematical Society Annual Meeting at ANU, Canberra. The talk was entitled “Strand algebras and contact categories”. Slides from the talk are available.
In October-November 2016 I gave two talks at the MSI Workshop on Low-Dimensional Topology & Quantum Algebraat ANU, Canberra. Some slides are available.
We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.
On 8 April, 2016 I gave a talk at the University of Melbourne in the Knot Invariants seminar. The talk was entitled “Hyperbolic volume and the Mahler measure of the A-polynomial”.
On 14 March, 2016 I gave a talk at Monash University in the Discrete Mathematics seminar. The talk was entitled “Trinities, hypergraphs, and contact structures”. Slides from the talk are available.
On 12 January, 2016 I gave a talk at the Workshop on Gromov-Witten theory, Gauge Theory, and Dualities at ANU Kioloa. The talk was entitled “Trinities, sutured Floer homology, and contact structure”. Slides from the talk are available.