  We consider topological defect networks with junctions in $A_{N-1}$ Toda CFT
and the connection to supersymmetric loop operators in $\mathcal{N} = 2$
theories of class S on a four-sphere. Correlation functions in the presence of
topological defect networks are computed by exploiting the monodromy of
conformal blocks, generalising the notion of a Verlinde operator. Concentrating
on a class of topological defects in $A_2$ Toda theory, we find that the
Verlinde operators generate an algebra whose structure is determined by a set
of generalised skein relations. These relations encode the representation
theory of a quantum group. In the second half of the paper, we explore the
dictionary between topological defect networks and supersymmetric loop
operators in the $\mathcal{N}=2^*$ star theory by comparing to exact
localisation computations. In this context, the the generalised skein relations
are related to the operator product expansion of loop operators.
