CAREER: New approaches to classical knot invariants

University Of Iowa, Iowa City IA

Investigators

Abstract

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.