Dehn twists on Kauffman bracket skein algebras

TL;DR

This paper provides an explicit formula for how Dehn twists act on the completed filtered Kauffman bracket skein modules of surfaces, advancing understanding of their algebraic and geometric structures.

Contribution

It introduces filtrations of the skein algebra and modules, enabling explicit computation of Dehn twist actions on these algebraic objects.

Findings

Explicit Dehn twist action formula derived

Filtrations of skein algebra introduced

Enhanced understanding of skein module transformations

Abstract

We give an explicit formula for the action of the Dehn twist along a simple closed curve in a compact connected oriented surface on the completion of the filtered skein modules. To do this, we introduce filtrations of the Kauffman bracket skein algebra and the Kauffman bracket skein modules on the surface.