3d Quantum Trace Map

TL;DR

This paper constructs a 3d quantum trace map linking skein modules and quantum gluing modules of 3-manifolds, extending 2d quantum trace concepts into three dimensions.

Contribution

It introduces the first explicit construction of the 3d quantum trace map, confirming a conjecture and generalizing the 2d quantum trace map to 3-dimensional topology.

Findings

Established the existence of the 3d quantum trace map.

Connected skein modules with quantum character varieties in 3D.

Extended the framework of quantum topology to three dimensions.

Abstract

We construct the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two quantizations of the character variety of ideally triangulated 3-manifolds. This map, whose existence was conjectured earlier by Agarwal, Gang, Lee, and Romo, is a natural 3-dimensional analog of the 2d quantum trace map of Bonahon and Wong. Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.