Homotopy Methods in Knot Theory

我建议使用的方法从代数拓扑学的研究结。 一个纽结诱导出一个从纽结上的构形空间到周围流形上的构形的映射。 利用最近发展起来的紧化技术，人们可以在这些构形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贝斯在德拉姆理论中以及PI和他的合作者在同伦理论本身中开创的。 Volic的结果可以用来证明所有的Bott-Taubes不变量，因而所有的真实的有限型不变量，都是这个诱导映射的同伦不变量。 已经在最低程度上，新的几何理解产生于研究诱导映射直接同伦，我建议继续在更高的程度上进行这项研究。 我还建议更深入地了解在这个理论中的作用的操作，以及扩展这些技术来研究链接homotopy.Knot理论，研究嵌入回路在空间中，是最古老和最杰出的领域之一拓扑。 在纽结理论的大部分时间里，它是与拓扑学的其他子领域并行发展的。 但在过去的二十年里，该领域已经从以前不相关的领域的影响发生了巨大的变化。 特别是量子场论提供了突破性的新结构。 人们可以尝试将结的能量定义为不变量，但为了做到这一点，必须“对所有连接进行积分”，也就是说，在结所处的空间上放置“能量场”的所有方式。 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，这意味着将其近似到有限阶。 拓扑学家在这种情况下使这种积分精确而严格，并证明它们为纽结的有限型不变量提供了基础。 但拓扑学家希望将该理论与拓扑学中更标准的结构重新连接起来。 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见解，因为最简单的量子不变量现在可以通过计算一条线在四个地方与一个结相交的实例来计算。 我希望在这项工作中既能找到与经典拓扑学的新联系，又能找到新颖的几何解释。

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.