Research in Knots, Links and 3-Manifolds

University Of California-Riverside, Riverside CA

Investigators

Abstract

Abstract Award: DMS-0102231 Principal Investigator: Xiao-Song Lin The central theme of this project is to explore as thoroughly as possible the significance of the Jones-Witten and Vassiliev invariants to knots, links, and 3-manifolds as topological structures. To be more specific, we will study the thermodynamic limit of the colored Jones polynomial using various probabilistic models; to explore beyond a simple formulation of the Casson invariant we found; to understand the congruence relation among Ohtsuki's invariants through congruence subgroups of the modular group; to find the normal form of a degree 1 map from a knot complement to another knot complement; to study the cohomology of knot complexes; to explore the applicability of classical techniques in 3-manifold topology to the study of Vassiliev invariants; and to continue the study of value and root distributions of the Jones polynomial. The phenomenon of knotting is a fundamental feature of the space that we live in, and knot theory is thus an important part of mankind's scientific knowledge because it is aimed at understanding the interplay of mathematical formulae and space structures. It is no wonder that concepts and tools originated from knot theory have been used in many areas of mathematics, as well as in chemistry, biology, physics, and computer science. For example, geneticists utilize knot theory to understand the processes of DNA replication and the function of enzymes that "unknot" DNA strands, and chemists use knot theory to understand and distinguish between different types of molecules. Our ability to discern different knots could well be served as a test of our scientific understanding of space structures.