On 3d Quantum Trace Maps

Abstract: A 3d quantum trace map is a homomorphism from the skein module of an ideally triangulated 3-manifold to its quantum gluing module that quantizes the classical trace map. There are two constructions of such maps, one by Garoufalidis and Yu in [GY], and the other by Panitch and Park in [PP1]. However, the relationship between these two constructions was unknown. We propose a third construction of the 3d quantum trace map which agrees with the one given by Garoufalidis and Yu, and extends to certain types of manifolds with ideally triangulated boundaries. Our 3d quantum trace map can be compared with that of [PP1] relatively easily by subdividing the face suspensions of [PP1] and the ideal tetrahedra of our definition into a common subdivision based on face cones. This allows us to give an exact relation between the definitions, which partially addresses the equivalence between the constructions of [GY] and [PP1].