CAREER: New approaches to classical knot invariants

许多最新的3流形拓扑技术都源于Heegaard分裂理论——将3流形分解成两个简单的部分。例如，Heegaard flower同源性被用来获得具有解结数1的结的障碍物。这样就可以对所有解结次数最多为10的1号结进行分类。与此同时，Cho和McCullough定义了隧道1号结树，该树给出了所有结的索引，其外部具有2 Heegaard分裂属。这些结的基本性质可以由结在树上的位置推断出来。PI目前正在开发新技术来研究闭合流形和结补中的表面。她建议将这些技术与上述结果结合起来，解决3流形拓扑中两个重要且长期存在的问题：建立结点交叉数的可加性，以及将Cho和McCulough的工作推广到所有结点。对于她的教育部分，她提出了一个项目，其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数学天才高中生，并培养他们的才能，最初通过为期两周的暑期项目，然后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-流形和结的结果常常让我们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。该领域的各种应用促进了该领域的研究和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、最简单的问题仍未得到解决。一个明显的例子是连接和下的结点交叉数的行为。交点数是结图中自交点的最小数量。一个多世纪以前，数学家提出了这样一个问题：如果两个结“融合”成一个结，那么这个结的交叉数与原来一对的交叉数相比是多少？在这个问题上几乎没有进展这一事实证明了这一领域的困难。通过将她的研究与她的教育工作结合起来，特别是在开发创新项目方面，通过招募和支持来自弱势背景的学生来解决未来数学家和科学家的短缺问题，PI工作的意义将大大增强。PI的一个主要目标是发现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，首先通过为期两周的暑期项目，然后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.