On quasi-morphisms from knot and braid invariants

TL;DR

This paper explores the construction of quasi-morphisms on pure braid groups and area-preserving diffeomorphisms using knot invariants like knot Floer and Khovanov homology, revealing new links between knot theory and symplectic geometry.

Contribution

It introduces a method to generate quasi-morphisms from knot invariants and discusses their application to diffeomorphism groups, extending the Gambaudo-Ghys construction.

Findings

Quasi-morphisms can be built from specific knot invariants.

Knot Floer and Khovanov homologies are used to construct these quasi-morphisms.

Potential applications to the study of diffeomorphism groups are discussed.

Abstract

We study quasi-morphisms on the groups Pn of pure braids on n strings and on the group D of compactly supported area-preserving diffeomorphisms of an open two-dimensional disc. We show that it is possible to build quasi-morphisms on Pn by using knot invariants which satisfy some special properties. In particular, we study quasi-morphisms which come from knot Floer homology and Khovanov-type homology. We then discuss possible variations of the Gambaudo-Ghys construction, using the above quasi-morphisms on Pn to build quasi-morphisms on the group D of diffeomorphisms of a 2-disc.