CAREER: New approaches to classical knot invariants

三维流形拓扑学中的许多最新技术都源于Heegaard分裂理论--将一个三维流形分解成两个简单的部分。例如，Heegaard Floer同调被用来获得具有解开数字1的结的障碍。这允许将所有解开的1号结的交叉数最多10个分类。同时，Cho和McCullough定义了隧道1号纽结的树，它给出了外部有亏格2 Heegaard分裂的所有纽结的索引。这些节点的基本属性可以通过节点在树中的位置来推断。PI目前正在开发研究闭流形和纽结补中的曲面的新技术。她建议利用这些技巧和前述结果来解决3-流形拓扑学中两个长期存在的重要问题：建立交叉点个数的可加性，并将Cho和Mcculough的工作推广到所有点。在她的教育部分，她提出了一个项目，主要目标是发现那些数学天赋不能代表他们潜力的高中生，并培养他们的才华，最初是通过一个为期两周的暑期计划，后来通过现有的支持系统。3-流形的拓扑结构是当前自然科学中许多感兴趣的问题的核心。关于3-流形和纽结的结果通常使我们能够更好地了解我们所生活的世界。蛋白质折叠、DNA打结和弦理论只是低维拓扑是关键研究工具的几个领域。这一领域的各种应用促进了该领域研究的增加和大量新成果和新技术的出现。尽管取得了这些进展，但许多最古老、最简单的国家问题仍然没有得到解决。这方面的一个明显的例子是在连接和下交叉节点数的行为。交叉数是结点图中自交的最小个数。一个多世纪前，数学家们提出了这样一个问题：如果两个纽结被“融合”成一个纽结，这个纽结的交叉数是多少？在这一问题上进展甚微的事实证明了这一领域的困难。通过将她的研究与她的教育努力相结合，PI的工作的意义将大大增强，特别是在开发创新项目方面，这些项目通过招收和支持来自代表性不足背景的学生来解决预测的未来数学家和科学家的短缺问题。PI的一个主要目标是找出那些在数学上有天赋的高中生，他们之前的数学成绩并不能代表他们的潜力，并培养他们的才华，最初是通过一个为期两周的暑期计划，后来是通过现有的支持系统。

Many recent techniques in 3-manifold topology have their roots in the theory of Heegaard splittings -- decompositions of a 3-manifold into two simple pieces. For example, Heegaard Floer homology was used to obtain obstructions to a knot having unknotting number one. This allowed for a classification of all unknotting number one knots of crossing number up to ten. At the same time, Cho and McCullough defined the tree of tunnel number one knots that gives an indexing of all knots whose exterior has a genus two Heegaard splitting. Fundamental properties of these knots can be deduced by where in the tree the knot is located. The PI is currently in the process of developing new techniques to study surfaces in closed manifolds and in knot complements. She proposes to use these techniques together with the aforementioned results to address two important and long-standing problems in 3-manifold topology: establishing the additivity of crossing number of knots, and generalizing the work of Cho and McCulough to all knots. For her educational component she proposes a program with a principal goal to identify mathematically-gifted high school students whose previous mathematical achievements do not represent their potential, and nurture their talents, initially through a two-week summer program and later via existing support systems.The topology of 3-manifolds is at the core of many of the current questions of interest in the natural sciences. Results about 3-manifolds and knots often allow us to gain better understanding of the world we live in. Protein folding, DNA knotting and string theory are just a few of the areas where low dimensional topology is a key research tool. The variety of applications of this field has served as a catalyst for increased research in the area and a plethora of new results and techniques. In spite of these developments, many of the oldest and simplest-to-state questions remain unsolved. A stark example of this is the behavior of crossing number of knots under connect sum. The crossing number is the minimum number of self-intersections in a diagram of the knot. More than a century ago mathematicians asked the following question: if two knots are "fused" into a single knot, what is the crossing number of this knot in terms of the crossing numbers of the original pair? The fact that little progress has been made on this question is evidence of the difficulty of this field. The significance of the work of the PI will be greatly enhanced by integrating her research with her educational efforts, particularly in terms of developing innovative programs that address forecasted future shortages of mathematicians and scientists by recruiting and supporting students from underrepresented backgrounds. A principal goal of the PI is to identify mathematically-gifted high school students whose previous mathematical achievements do not represent their potential, and to nurture their talents, initially through a two-week summer program and later via existing support systems.