Knots and 3-Manifolds

University Of California-Davis, Davis CA

Investigators

Abstract

Abstract Award: DMS-0104126 Principal Investigator: Abigail A. Thompson The focus of the proposed research is the study of knots and 3-dimensional spaces. We will examine questions about surgery on knots, knotting of graphs in 3-space dimensional spaces, and Heegaard decompositions of 3-manifolds. We describe three specific goals and their connections to some of the basic questions in the field. The first goal is to develop the tools and techniques of multi-parameter thin position and sweep-outs so that they can be used to tackle some of the long-standing problems in knot theory and 3-manifold theory. A second is to extend work of the proposer and M. Scharlemann on unknotted planar graphs in the 3-sphere to a more general setting. Finally, we aim to obtain a clearer understanding of general questions about 3-manifolds by obtaining a deeper understanding of 3-manifolds of Heegaard genus two. We will examine specific questions about tunnel number one knots, special cases of larger questions about what genus two manifolds can be obtained by surgery on a knot in the 3-sphere, which of these manifolds contain immersed surfaces with injective fundamental groups, and how information from the Heegaard diagrams of these manifolds translates into geometric and algebraic information about the manifold itself. We live in what is apparently a universe with three spatial dimensions. Exploring the possible forms that our universe might have is not only a deep problem for physicists but also for mathematicians. Different mathematical possibilities lead to very different expected physical properties. As an example, we do do not know if, were we able to program a very fast rocket to go "straight" into space, it would eventually return to its starting point, like a ship on the surface of the earth, or if it would continue to travel away from us forever. There are an infinite number of possibilities for the shape of the universe. This project proposes to explore some of them using the techniques of low-dimensional topology and knot theory.