Cut-and-paste conjectures and multicurves

Introduction

Knot theory has seen extraordinary developments over the past decades. The arrival of modern homological knot invariants has had far-reaching implications beyond low-dimensional topology, giving insight into old problems through deep ties between knot theory, algebraic geometry, representation theory, Floer theory, and physics.

Project Aim

My ERC project aims to establish a new perspective on knot homology theories using a new type of invariants, so-called multicurves. As objects of Fukaya categories of simple surfaces, these multicurve invariants make local versions of knot homology theories amenable to essentially combinatorial techniques.

Geometric Properties

Thanks to their exceptional geometric and gluing properties, multicurves are ideally suited to implement the divide-and-conquer principle for attacking hard open problems. In fact, I have not only been directly involved in the definition of three of these invariants, but I have also applied them to resolve several open conjectures in the field already.

Research Programme

The purpose of my research programme is to investigate fundamental open problems in low-dimensional topology that require a deeper understanding of the new technology of multicurves. To this end, I will pursue the following four lines of basic research:

1. Investigate the topological properties of the new invariants and their relation to classical invariants.
2. Explore the existence of local versions of various spectral sequences that are known to relate knot homology theories.
3. Make the invariants more computable.
4. Apply the generic principles that underlie the definition of multicurves to other settings.