HOMOLOGY THEORIES FOR TANGLES AND BORDERED 3-MANIFOLDS

Heegaard花同调是3-流形及其结的一个新的、强大的不变量。在许多其他的好处中，它检测结属（最不复杂的表面，其边界是结），特别是，结是否是非平凡的。它是一个同调理论，像许多同调理论一样，欧拉特征很有趣：对于结，欧拉特征是最古老的结不变量之一，亚历山大多项式。Heegaard flower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结多项式。但是它很难计算，并且是由特别规则定义的，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 flower同调和其他同调理论推广到有边界的对象（结点或3-流形），这样当一个结点或流形被分割成碎片时，整体的不变量可以由碎片的不变量计算出来。在其他好处中，这将使理论更易于计算，并为理论提供公理。尽管人们对结理论的研究已经有好几个世纪了，但一些最基本的问题，如寻找边界位于结上的最不复杂的曲面，仍然不容易回答。heeggaard flower同调是一个最新的理论，它回答了这个问题以及结理论和拓扑学中的许多其他问题。然而，即使是相对较小的结点也很难计算。在这个项目中，我和我的合作者将扩展Heegaard flower同源性（以及其他相关理论），以便它们可以计算结的片段，然后构建完整的结。这有望提供一个伟大的计算和理论工具。在整个项目中，可访问性将被强调，并且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.