CAREER: New approaches to classical knot invariants

许多最近的技术在3-流形拓扑有其根源的理论Heegaard分裂-分解的3-流形成两个简单的部分。例如，使用Heegaard Floer同源性来获得对具有解结编号1的结的障碍物。这允许分类的所有unknotting号码一个结的交叉数高达十。与此同时，Cho和McCullough定义了隧道一号节点的树，该树给出了外部具有亏格2 Heegaard分裂的所有节点的索引。这些节点的基本性质可以通过节点在树中的位置来推断。PI目前正在开发新的技术来研究封闭流形和结补中的曲面。她建议使用这些技术与上述结果一起解决3-流形拓扑中两个重要且长期存在的问题：建立节点交叉数的可加性，并将Cho和McCulough的工作推广到所有节点。对于她的教育部分，她提出了一个计划，其主要目标是识别具有数学天赋的高中生，他们以前的数学成就并不代表他们的潜力，并培养他们的才能，最初通过为期两周的暑期课程，后来通过现有的支持系统。三维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。关于三维流形和纽结的结果常常使我们更好地理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学作为关键研究工具的几个领域。这一领域的各种应用已经成为该领域研究增加的催化剂，并产生了大量的新成果和新技术。尽管取得了这些进展，但许多最古老和最简单的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。交叉数是纽结图中自交的最小数目。世纪以前，数学家们提出了这样一个问题：如果两个结“融合”成一个结，这个结的交叉数与原来的交叉数之比是多少？在这个问题上进展甚微，这一事实证明了这一领域的困难。PI的工作的重要性将大大提高她的研究与她的教育工作相结合，特别是在开发创新计划，通过招募和支持来自代表性不足的背景的学生来解决预测未来数学家和科学家短缺方面。PI的一个主要目标是识别那些以前的数学成绩不能代表他们潜力的有数学天赋的高中生，并培养他们的才能，最初通过为期两周的暑期课程，后来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.