Homotopy Methods in Knot Theory

我建议使用代数拓扑学中的方法来研究纽结。纽结诱导出从纽结上的构形空间到环境流形中的构形的映射。利用最近发展的紧化技术，人们可以在这些配置空间上固定边界条件，并研究这个诱导映射相对于这些边界条件的同伦类。这种方法是由博特和陶布斯在德拉姆理论中率先提出的，也是由PI和他的合作者在同伦理论中提出的。Volic的结果可以用来证明所有的Bott-Taube不变量，从而所有的实有限类型不变量，都是这个诱导映射的同伦不变量。对诱导映射的直接同伦的研究已经在最低程度上产生了新的几何认识，我建议在更高的程度上继续这一研究。我还建议更深入地理解歌剧在这一理论中的作用，并将这些技术扩展到研究链同伦。纽结理论是拓扑学中最古老和最杰出的领域之一，它研究空间中的嵌入环。在其生命的大部分时间里，纽结理论一直与拓扑学的其他子领域并行发展。但在过去的二十年里，这个领域已经从以前不相关的领域的影响中发生了戏剧性的变化。特别是，量子场论提供了开创性的新结构。一个人可以尝试将一个结的能量定义为一个不变量，但为了这样做，一个人必须“整合所有的连接”，也就是说，通过所有的方式在这个结所在的空间上放置一个“能量场”。这样的积分并不以精确的数学形式存在，但通过量子场论中的标准微扰展开，人们可以写下费曼积分，这些积分旨在将其近似到有限阶。拓扑学家已经在这种情况下使这种积分精确而严谨，并表明它们为纽结的有限类型不变量提供了基础。但拓扑学家希望将这一理论与拓扑学中更标准的结构重新联系起来。PI之前的工作标志着这种联系的开始。在这个过程中获得了新的几何洞察力，因为现在最简单的量子不变量可以通过计算一条线在四个位置上与一个结相交的实例来计算。我希望在这项拟议的工作中既能找到与经典拓扑学的新联系，也能找到新的几何解释。

I propose to use methods from algebraic topology in the study of knots. A knot induces a map from the space of configurations on the knot to configurations in the ambient manifold. With recently developed compactification technology, one can fix boundary conditions on those configuration spaces and study the homotopy class of this induced map relative to those boundary conditions. Such an approach was pioneered by Bott and Taubes in de Rham theory and by the PI and his collaborators in homotopy theory itself. Results of Volic may be used to show that all Bott-Taubes invariants, and thus all real finite-type invariants, are homotopy invariants of this induced map. Already in lowest degree, new geometric understanding arises from studying the induced map directly homotopy, and I propose to continue this study in higher degrees. I also propose to more deeply understand the role of operads in this theory, as well as to extend these techniques to study link homotopy.Knot theory, the study of embedded loops in space, is one of the oldest and most distinguished fields in topology. For most of its life, knot theory has developed in parallel with other subfields of topology. But in the last twenty years, the field has changed dramatically from the influence of previously unrelated fields. In particular, quantum field theory has provided ground-breaking new constructions. One can try to define the energy of a knot as an invariant, but in order to do so one must "integrate over all connections", that is over all ways of putting an "energy field" on the space in which the knot lives. Such an integral does not exist in precise mathematical form, but through the standard perturbative expansion in quantum field theory one can write down Feynman integrals which are meant to approximate it to finite order. Topologists have made such integrals precise and rigorous in this setting, and shown that they provide a basis for the finite-type invariants of knots. But topologists would like to reconnect the theory to more standard constructions in topology. The PI's previous work marks the beginning of such a connection. New geometric insight was gained in the process, as the simplest quantum invariant can now be computed by counting instances of a line intersecting a knot in exactly four places. I hope to find both new connections with classical topology and novel geometric interpretations in this proposed work.