  Non-perturbative aspects of $\mathcal{N}=2$ supersymmetric gauge theories of
class $\mathcal{S}$ are deeply encoded in the algebra of functions on the
moduli space $\mathcal{M}_\text{flat}$ of flat $SL(N)$-connections on Riemann
surfaces. Expectation values of Wilson and 't Hooft line operators are related
to holonomies of flat connections, and expectation values of line operators in
the low-energy effective theory are related to Fock-Goncharov coordinates on
$\mathcal{M}_\text{flat}$. Via the decomposition of UV line operators into IR
line operators, we determine their noncommutative algebra from the quantization
of Fock-Goncharov Laurent polynomials, and find that it coincides with the
skein algebra studied in the context of Chern-Simons theory. Another
realization of the skein algebra is generated by Verlinde network operators in
Toda field theory. Comparing the spectra of these two realizations provides
non-trivial support for their equivalence. Our results can be viewed as
evidence for the generalization of the AGT correspondence to higher-rank class
$\mathcal{S}$ theories.
