Triangular decomposition of skein algebras

TL;DR

This paper introduces a refined skein algebra for surfaces, decomposes it into triangle-based blocks, and simplifies the proof of the quantum trace map's existence, connecting it with Muller’s skein algebra.

Contribution

It presents a finer skein algebra decomposition aligned with ideal triangulations and simplifies the quantum trace map proof, extending it to Muller’s skein algebra.

Findings

Decomposition of skein algebra into triangle blocks

Simplified proof of quantum trace map existence

Extension of quantum trace map to Muller skein algebra

Abstract

By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.