Noncommutative Techniques in Knot Theory

纽结理论的一个基本目标是将纽结分类到合痕和和谐。纽结补的基本群是一个重要的代数不变量，但很少是交换群。具有局部系数的同调可以用于将模块与结相关联。由于模是阿贝尔群，区分它们仍然是可行的。然而，由于这些通常是非交换环上的模，基本群的一些丰富结构被保留下来。这些同调模，以及在它们上面定义的连接形式，已经被我和其他人用来定义纽结的合痕不变量和纽结的和谐不变量。这个项目涉及在纽结理论的两个领域使用这些非交换技术：纽结一致性和纽结弗洛尔同源性。这个项目的目标是使用非交换代数更好地理解三维和四维空间中的纽结曲线。结的研究在生物学和物理学中有重要的应用。例如，DNA链是自然打结的，但必须解开才能复制。许多数学家使用代数来更好地理解曲线如何打结，然而所考虑的代数类型通常是交换的。通过使用非交换代数，我们可以更精确地理解曲线如何在三维空间中纽结，以及如何在四维空间中结合曲面。

A fundamental goal in knot theory is the classification of knots up to isotopy and up to concordance. The fundamental group of the knot complement is an important algebraic invariant, but is rarely abelian. Homology with local coefficients can be used to associate modules to the knot. Since modules are abelian groups, distinguishing them is still feasible. However, since these are often modules over noncommutative rings, some of the rich structure of the fundamental group is retained. These homology modules, as well as linking forms defined on them, have been used by myself and others to define isotopy invariants of knots and concordance invariants of knots. This project concerns using these noncommutative techniques in two areas of knot theory: knot concordance and knot Floer homology.The goal of this project is to use non-commutative algebra to better understand knotted curves in 3- and 4-dimensional space. The study of knots has important applications in biology and physics. For example, DNA strands are naturally knotted, but must unknot in order to replicate. Many mathematicians have used algebra to better understand how curves can knot, however the type of algebra considered is usually commutative. By employing non-commutative algebras, we can get a more refined understanding for how curves can knot in 3-dimensional space and bound surfaces in 4-dimensional space.