3d quantum trace map

Panitch, Samuel; Park, Sunghyuk

doi:10.2140/agt.2026.26.2255

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$3$d quantum trace map

Samuel Panitch and Sunghyuk Park

Algebraic & Geometric Topology 26 (2026) 2255–2314

DOI: 10.2140/agt.2026.26.2255

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, Romo (2022), is a natural 3-dimensional analog of the 2d quantum trace map of Bonahon and Wong (2011). Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.

Keywords

quantum trace map, skein module, quantum Teichmüller space, quantum gluing module

Mathematical Subject Classification

Primary: 57K31

References

Publication

Received: 29 July 2024

Revised: 16 June 2025

Accepted: 30 June 2025

Published: 22 June 2026

Authors

Samuel Panitch
Department of Mathematics Yale University New Haven, CT United States

Sunghyuk Park
Department of Mathematics Harvard University Cambridge, MA United States

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Volume 26, issue 6 (2026)