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Abstract We consider the quantum UV-IR map for line defects in class S theories of gl $$ \mathfrak{gl} $$ (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 N $$ \mathcal{N} $$ = 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 ℝ3, as a sum over BPS webs attached to the link.