An Enumerative Approach to Heegaard Floer

如何区分三角形和六边形？简单：计算边E或顶点v。事实上，这两个“不变量”是等价的，因为它们由方程X=V-E+F联系在一起，其中F计算由边勾勒的二维表面或“面”的数量，而X是一个被称为欧拉特征的百年不变量。n-gon具有欧拉特征X=n-n+1=1，而足球表面具有欧拉特征X=60-90+32=2。事实上，任何二维球体都有X=2，无论它是如何“绗缝”的。裹着百吉饼的被子具有欧拉特性0，有g孔的椒盐卷饼表面具有X=2-2G。另一个不变量，“基本群”，在佩雷尔曼对庞加莱猜想的证明和阿戈尔的突破奖研究之间，最近成为头条新闻。基本组跟踪在对象周围缠绕字符串的可能方法。绳子只是从足球上掉下来，但它可以在百吉饼上缠绕无数次，或者无望地缠在椒盐卷饼上。数学家在几何学和拓扑学方面的许多工作都涉及类似的“不变量”。上面的绗缝表面被称为2-流形，因为是由二维碎片构成的。我研究了一个3流形的不变量Heegaard flower同调（HF），它的发明是为了帮助计算一个4流形不变量，以区分在4流形上进行微积分的不同自洽方法。虽然HF已经有近20年的历史了，但在许多情况下，HF应该告诉我们关于这些3流形的什么仍然是一个谜。然而，最近的一些猜想可能为解开这个谜团提供了钥匙。Boyer， Gordon和Watson已经推测出具有消失HF的3-流形（“l空间”）和3-流形之间的二分法，其中基本群中的所有包膜都可以被赋予“左序”(LO)-一种在群上的特殊序，它与额外的弦包膜的添加是自一致的。与此同时，Juhasz推测了l空间和具有共取向紧叶理（CTF）的3-流形之间的二分法——一种用平面（如纸张）雕刻3-流形和具有负欧拉特征的2-流形的特定方式。Nemethi最近证明了一个关于l空间和代数奇点的定理——由多项式之间的关系控制的特殊压扁点。受到这些观察的启发，我自己的研究努力超越这种二分法，而是使用HF作为区分和计数LOs， CTFs和某些其他结构的工具。这种方法以前从未被采用过，但是我已经开发了新的工具，并且已经开始交付结果。这些努力中的任何一项的成功都可能彻底改变几何学家和拓扑学家能够回答的问题类型。事实上，令人惊讶的是，尽管错综复杂的物体的行为，特别是在分子水平上，影响着我们的日常生活，但数学家对我们所居住的世界的三维几何和拓扑所知甚少。例如，打结DNA的拓扑结构影响拓扑异构酶制备DNA转录的速度，这是某些癌症治疗发展中的一个重要因素。帝国理工学院（Imperial College）的多萝西·巴克（Dorothy Buck）博士研究了这些问题，她的许多学生使用Heegaard flower同源性来推进我们对结和缠结的三维拓扑结构的理解。谁知道氢氟酸未来会有什么应用？

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?