  We propose a method for determining the spins of BPS states supported on line
defects in 4d $\mathcal{N}=2$ theories of class S. Via the 2d-4d
correspondence, this translates to the construction of quantum holonomies on a
punctured Riemann surface $\mathcal{C}$. Our approach combines the technology
of spectral networks, which decomposes flat $GL(K,\mathbb{C})$-connections on
$\mathcal{C}$ in terms of flat abelian connections on a $K$-fold cover of
$\mathcal{C}$, and the skein algebra in the 3-manifold $\mathcal{C}\times
[0,1]$, which expresses the representation theory of the quantum group
$U_q(gl_K)$. With any path on $\mathcal{C}$, the quantum holonomy associates a
positive Laurent polynomial in the quantized Fock-Goncharov coordinates of
higher Teichm\"uller space. This confirms various positivity conjectures in
physics and mathematics.
