On 12 May, 2014 I gave a talk at the Monash University Discrete Mathematics Seminar. Slides are available.
An explicit formula for the A-polynomial of twist knots
We extend Hoste-Shanahan’s calculations for the A-polynomial of twist knots, to give an explicit formula.
Twisty itsy bitsy topological field theory
We extend the topological field theory (“itsy bitsy topological field theory”‘) of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (“itsy”) and quantum-field-theoretic (“bitsy”) aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.
A Yang-Baxter equation from sutured Floer homology, Sep 2013
On 30 September, 2013 I gave a talk at the 57th Australian Mathematical Society Annual Meeting, at the University of Sydney, as part of the Geometry and Topology session. Slides are available.
Sutures, quantum groups, TQFT, May 2013
On 24 May, 2013 I gave a talk at the University of Melbourne, for the Algebra-Geometry-Topology Seminar. The talk was entitled “Sutures, quantum groups and topological quantum field theory”. Slides are available.
Contact topology and elementary combinatorics, April 2013
On 16 April, 2013 I gave a talk at ANU, for the Algebra and Topology seminar. The talk was entitled “Contact topology and holomorphic invariants via elementary combinatorics”. Slides are available.
Contact topology and holomorphic invariants via elementary combinatorics
In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3-manifolds such as Heegaard Floer homology and embedded contact homology. These invariants are based on holomorphic curves and moduli spaces, but in the simplest cases, some of their structure reduces to some elementary combinatorics and algebra which may be of interest in its own right. In this note, which is essentially a light-hearted exposition of some previous work of the author, we give a brief introduction to some of the ideas of contact topology and holomorphic curves, discuss some of these elementary results, and indicate how they arise from holomorphic invariants.
Contact topology and holomorphic invariants via elementary combinatorics, Monash Dec 2012
On 7 December, 2012 I gave a talk at Monash University. Slides are available.
Field-theoretic ideas from contact geometry, ANZAMP 2012
On 5 December, 2012 I gave a talk at the Australian and New Zealand Association of Mathematical Physics (ANZAMP) Inaugural annual meeting. The talk was entitled “Some field-theoretic ideas out of contact geometry and elementary topology”. Slides are available.
Dimensionally-reduced sutured Floer homology as a string homology
Joint with Eric Schoenfeld.
We show that the sutured Floer homology of a sutured 3-manifold of the form [latex](D^2 \times S^1, F \times S^1)[/latex] can be expressed as the homology of a string-type complex, generated by certain sets of curves on [latex](D^2, F)[/latex] and with a differential given by resolving crossings. We also give some generalisations of this isomorphism, computing “hat” and “infinity” versions of this string homology. In addition to giving interesting elementary facts about the algebra of curves on surfaces, these isomorphisms are inspired by, and establish further, connections between invariants from Floer homology and string topology.