  The $q$-deformed loop gravity framework was introduced as a canonical
formalism for the Turaev-Viro model (with $\Lambda < 0$), allowing to quantize
3D Euclidean gravity with a (negative) cosmological constant using a quantum
deformation of the gauge group. We describe its application to the 2-torus,
explicitly writing the $q$-deformed gauge symmetries and deriving the reduced
physical phase space of Dirac observables, which leads back to the Goldman
brackets for the moduli space of flat connections. Furthermore it turns out
that the $q$-deformed loop gravity can be derived through a gauge fixing from
the Fock-Rosly bracket, which provides an explicit link between loop quantum
gravity (for $q$ real) and the combinatorial quantization of 3d gravity as a
Chern-Simons theory with non-vanishing cosmological constant $\Lambda<0$. A
side-product is the reformulation of the loop quantum gravity phase space for
vanishing cosmological constant $\Lambda=0$, based on $\mathrm{SU}(2)$
holonomies and $\mathfrak{su}(2)$ fluxes, in terms of $\mathrm{ISU}(2)$
Poincar\'e holonomies. Although we focus on the case of the torus as an
example, our results outline the general equivalence between 3D $q$-deformed
loop quantum gravity and the combinatorial quantization of Chern-Simons theory
for arbitrary graph and topology.
