CAREER: Extending and unifying modern homological invariants in low dimensional topology

Princeton University, Princeton NJ

Investigators

Abstract

The project will concentrate on two modern homological knot invariants: Khovanov homology and knot Floer homology. Both invariants associate chain complexes to knots whose chain homotopy types (and consequently, homology groups) are knot invariants. This project has two major research goals. The first goal is to extend various aspects of these homological invariants to stable homotopy types, i.e., construct new knot invariant topological spaces (well-defined up to stable homotopy equivalences) whose homology groups are the existing invariants. This will produce higher structures on the homological invariants which will be useful for studying certain geometric properties of knots, such as their four-ball genus. The second goal of this project is to study the relationship between the homological invariants. The project seeks to find new spectral sequences, and combinatorial reformulations of the existing spectral sequences, from the Khovanov homology invariants to the knot Floer homology invariants. This will lead to a better understanding of why these two invariants coming from very different origins share so many similarities. Topology is the branch of mathematics that studies shapes of spaces; and low dimensional topology concentrates on spaces up to dimension four. Knot theory is an important sub-field of low dimensional topology where one studies one-dimensional objects inside three-dimensional spaces, for example, knotted pieces of strings inside the (three-dimensional Euclidean) space that we live in. In addition to being an extremely valuable tool in low dimensional topology, knot theory has also proven to be useful in real world applications: from analysing knotting in DNA to studying mixing in liquids, and from estimating energies of orbits inside a magnetic field to creating new data encryption schemes. A fundamental problem in knot theory is the knot isotopy problem: to determine if a given knot can transform into another one without tearing or crossing itself (such a transformation is called a knot isotopy). Knot invariants are mathematical objects, such as numbers or groups, that one associates to knots, and which remain unchanged during such a knot isotopy. Therefore, knot invariants are used extensively in the knot isotopy problem: if one finds some knot invariant that takes different values on the two given knots, then one concludes that the two knots are not isotopic. The current project is based on knot theory, and it seeks to study properties of certain previously known knot invariants, and to extend them to construct new knot invariants. The project will lead to dispersion of mathematical knowledge, particularly in the area of low-dimensional topology, via a variety of means. This project will fund undergraduate students for summer research, week-long workshops on low-dimensional topology, and a wiki-based website on knot theory.