HOMOLOGY THEORIES FOR TANGLES AND BORDERED 3-MANIFOLDS

Heegaard Floer同调是其中的三维流形和纽结的一个新的强大的不变量。在许多优点中，它可以检测纽结亏格(边界是纽结的最不复杂的曲面)，特别是纽结是否是非平凡的。它是一个同调理论，和许多同调理论一样，欧拉特性很有趣：对于纽结，欧拉特性是最古老的纽结不变量之一，亚历山大多项式。Heegaard Floer同调是最近的一族同调理论的一部分，它的Euler特性给出了其他的纽结多项式。但它可能很难计算，而且是由临时规则定义的，而不是一组简明的属性。在这个项目中，我们将发展一种边界同调不变量理论：将Heegaard Floer同调和其他同调理论推广到有边界的对象(纽结或3-流形)，这样当一个纽结或流形被分割成碎片时，可以从碎片的不变量计算出整体的不变量。除了其他好处外，这将使该理论更易于计算，并为该理论提供公理。尽管纽结理论已经研究了许多世纪，但一些最基本的问题，如寻找边界位于纽结上的最不复杂的曲面，仍然不容易回答。Heegaard Floer同调是最近的一个理论，它回答了纽结理论和拓扑学中的许多其他问题。然而，即使对于相对较小的节点，也很难计算。在这个项目中，我和我的合作者将扩展Heegaard Floer同调(以及其他相关理论)，以便可以计算出纽结的片段，然后建立起完整的纽结。这有望提供一个很好的计算和理论工具。在整个项目中，将强调可访问性，并且该项目将与例如本科生研究计划相结合。

Heegaard Floer homology is a new and powerful invariant of 3-manifolds and knots in them. Among many other benefits, it detects the knot genus (the least complicated surface whose boundary is the knot), and in particular, whether a knot is non-trivial. It is a homology theory, and like many homology theories the Euler characteristic is interesting: for knots, the Euler characteristic is one of the oldest knot invariants, the Alexander polynomial. Heegaard Floer homology is part of a family of recent homology theories whose Euler characteristic gives other knot polynomials. But it can be hard to compute, and is defined by ad-hoc rules rather than a concise set of properties. In this project, we will develop a theory of bordered homology invariants: extend the Heegaard Floer homology and other homology theories to objects (knots or 3-manifolds) with boundary, so that when a knot or manifold is split into pieces the invariant for the whole can be computed from the invariants for the pieces. Among other benefits, this will make the theory more computable, and give axioms for the theory.Although knot theory has been studied for many centuries, some of the most elementary questions, such as finding the least complicated surface whose boundary lies on the knot, are still not easy to answer. Heegaard Floer homology is one recent theory that answers this and many other questions in knot theory and topology. However, it is hard to compute even for relatively small knots. In this project, I and my collaborators will extend Heegaard Floer homology (and other related theories) so that they can be computed for pieces of a knot and then built up to the complete knot. This promises to provide a great computational and theoretical tool. Throughout the project, accessibility will be emphasized, and the project will be integrated with, for instance, an undergraduate research program.