Low-dimensional topology; knot concordance and geography

Indiana University, Bloomington IN

Investigators

Abstract

Knots in 3-space are called concordant if they cobound an embedded annulus in the product of 3-space with a closed interval. The set of concordance classes forms an abelian group; this project investigates the structure of this concordance group. There are three major facets to the project: (1) Developing a better understanding of known concordance invariants and discovering new invariants, (2) Applying these invariants to classify important families of knots, and (3) Using these results to unravel the underlying structure of the concordance group. The perspective of classical physics is three-dimensional, focusing on the three spatial dimensions in which we live. Modern physics has a higher-dimensional perspective, for instance including time to create a four-dimensional model of the universe. In the 1960s it was recognized that four-dimensional spaces can be built by gluing together simpler pieces; the way these pieces are glued together can be represented by knots, with the complexity of the resulting space reflected in the complexity of the knots that occur. This proposal is focused on understanding knotting from a four-dimensional perspective. In particular, new tools will be developed, and these will be applied to complete our four-dimensional analysis of low- crossing number knots.