HOMOLOGY THEORIES FOR TANGLES AND BORDERED 3-MANIFOLDS

Heegaard Floer同调是三维流形及其纽结的一个新的、强有力的不变量。 在许多其他好处中，它检测结属（边界是结的最不复杂的表面），特别是，结是否是非平凡的。 这是一个同调理论，和许多同调理论一样，欧拉特征是有趣的：对于纽结，欧拉特征是最古老的纽结不变量之一，亚历山大多项式。 Heegaard Floer同调是最近同调理论家族的一部分，其欧拉特征给出了其他纽结多项式。 但它可能很难计算，并且是由特定规则定义的，而不是一组简洁的属性。 在这个项目中，我们将发展一个有边界的同调不变量理论：将Heegaard Floer同调和其他同调理论扩展到有边界的对象（结或3-流形），这样当一个结或流形被分割成碎片时，整体的不变量可以从碎片的不变量计算出来。 尽管纽结理论已经被研究了很多个世纪，但是一些最基本的问题，比如找到边界位于纽结上的最不复杂的曲面，仍然不容易回答。Heegaard Floer同调是一个最近的理论，它回答了纽结理论和拓扑学中的这个问题和许多其他问题。 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（和其他相关理论），以便它们可以计算一个结的片段，然后建立完整的结。 这有望提供一个伟大的计算和理论工具。 在整个项目中，将强调可访问性，该项目将与例如本科研究计划相结合。

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.