The quantum UV-IR map for line defects in $$ \mathfrak{gl} $$(3)-type class S theories

Abstract We consider the quantum UV-IR map for line defects in class S theories of $$ \mathfrak{gl} $$ 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 $$ \mathcal{N} $$ 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.