We consider the quantum UV-IR map for line defects in class $S$ theories of $\mathfrak{ gl}(3)$-type. This map computes the protected spin character which counts framed BPS states with spin for the bulk-defect system. We give a geometric method of computing this map motivated by the physics of five-dimensional $\mathcal{ N}=2$ supersymmetric Yang-Mills theory, and compute it explicitly in various examples. As a spin-off we propose a new way of computing a certain specialization of the HOMFLY polynomial for links in $\mathbb{ R}^3$, as a sum over BPS webs attached to the link.
Categories: hep-th math.GT math.QA math.RT