Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: the case $n=3$

Daniel C. Douglas

Published 2021-01-18Version 1

We generalize Bonahon and Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thickened punctured surface $\mathfrak{S} \times (0, 1)$ a Laurent polynomial $\mathrm{Tr}_\lambda^q(K) = \mathrm{Tr}_\lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm\"{u}ller space, depending on the choice of an ideal triangulation $\lambda$ of the surface $\mathfrak{S}$. Along the way, we propose a definition for a $\mathrm{SL}_n(\mathbb{C})$-version of this invariant.