Generalized Dehn twists in low-dimensional topology

TL;DR

This paper explores the algebraic and geometric properties of generalized Dehn twists in low-dimensional topology, including their realizability, diagrammatic descriptions, and relations to 3D cobordisms and skein algebras.

Contribution

It provides a comprehensive review of generalized Dehn twists, their algebraic definitions, realizability as surface diffeomorphisms, and their applications in 3D topology and skein algebra variants.

Findings

Generalized Dehn twists can be described via decorated trees and Hopf algebras.

They are realizable as diffeomorphisms in certain cases.

Connections to 3D homology cobordisms and skein algebra variants are established.

Abstract

The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going to the dimension three, we also overview the relation between generalized Dehn twists and $3$-dimensional homology cobordisms, and we survey the variants of generalized Dehn twists for skein algebras of the surface.