An Enumerative Approach to Heegaard Floer

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How do you distinguish a triangle from a hexagon? Easy: you count the edges E or vertices V. In fact, these two "invariants" are equivalent, since they are related by the equation X=V-E+F, where F counts the number of two dimensional surfaces or "faces" outlined by edges, and X is a centuries-old invariant called the Euler characteristic. An n-gon has Euler characteristic X=n-n+1=1, while the surface of a football has Euler characteristic X=60-90+32=2. In fact, any 2-dimensional sphere has X=2, no matter how it is "quilted." A quilt wrapping a bagel has Euler characteristic 0, and the surface of a G-holed pretzel has X=2-2G. Another invariant, the "fundamental group," has made recent headlines, between Perelman's proof of the Poincare conjecture and Agol's Breakthrough Prize research. The fundamental group keeps track of possible ways to wrap string around an object. The string just falls off a football, but it can wrap countably many times around a bagel or get hopelessly tangled on a pretzel.A lot of the work of mathematicians in geometry and topology deals with "invariants" analogous to these. The quilted surfaces above are called 2-manifolds, since built from 2-dimensional pieces. I study an invariant for 3-manifolds called Heegaard Floer homology (HF), which was invented to help calculate a 4-manifold invariant that distinguishes different self-consistent ways of doing calculus on 4-manifolds. Although HF is nearly 2 decades old, in many cases it remains a mystery what HF should tell us about these 3-manifolds.There are some recent conjectures, however, which might provide keys to unlock this mystery. Boyer, Gordon, and Watson have conjectured a dichotomy between 3-manifolds with vanishing HF--"L-spaces"--and 3-manifolds for which all the wrappings in the fundamental group can be given an "left order" (LO)--a special ordering on the group which is self-consistent with respect to the addition of additional string wraps. Juhasz, meanwhile, has conjectured a dichotomy between L-spaces and 3-manifolds with a co-oriented taut foliation (CTF)--a certain way of sculpting a 3-manifold out of planes (like sheets of paper) and 2-manifolds with negative Euler characteristic. Nemethi has recently proven a theorem relating L-spaces to algebraic singularities--special squashed up points governed by relationships among polynomials.Inspired by these observations, my own research strives to move beyond such dichotomies and instead use HF as a tool to distinguish and count LOs, CTFs, and certain other structures. This approach has never been taken before, but I have already developed new tools which have begun to deliver results. Success in any one of these endeavours could revolutionize the types of questions geometers and topologists are able to answer.In fact, it is astonishing how little mathematicians still know about the 3-dimensional geometry and topology of the world we inhabit, even though the behaviour of intricately tangled objects, particularly at the molecular level, impacts our daily lives. For example, the topology of knotted DNA influences the speed with which topoisomerase enzymes can prepare DNA for transcription, an important factor for certain cancer treatments in development. Dr Dorothy Buck at Imperial College studies questions such as these, and many of her students use Heegaard Floer homology to advance our understanding of the 3-dimensional topology of knots and tangles. Who knows what applications of HF the future might hold?