Homotopy Methods in Knot Theory

我建议在节的研究中使用代数拓扑的方法。一个结点诱导出一个从结点上的构型空间到周围流形中的构型的映射。利用最近发展的紧化技术，可以在这些构形空间上固定边界条件，并研究与这些边界条件相关的诱导映射的同伦类。这种方法是由博特和陶博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bot - taubes不变量，从而所有的实有限型不变量，都是这个诱导映射的同伦不变量。在最低程度上，通过直接研究诱导映射同伦产生了新的几何认识，我建议在更高程度上继续这项研究。我还建议更深入地理解operad在这一理论中的作用，并将这些技术扩展到链接同伦的研究中。结理论是对空间内嵌环的研究，是拓扑学中最古老和最杰出的领域之一。在其生命的大部分时间里，结理论与拓扑学的其他子领域并行发展。但是在过去的二十年里，这个领域从以前不相关的领域的影响中发生了巨大的变化。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，人们必须“对所有连接进行整合”，即在结所在的空间中放置一个“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，人们可以写出费曼积分，这意味着将其近似为有限阶。拓扑学家在这种情况下使这种积分精确而严格，并表明它们为结点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，人们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恰好四个地方与一个结相交的实例来计算。我希望在这个提议的工作中找到与经典拓扑的新联系和新的几何解释。

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.