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The inverse square potential arises in a variety of different quantumphenomena, yet notoriously it must be handled with care: it suffers frompathologies rooted in the mathematical foundations of quantum mechanics. Weshow that its recently studied conformality-breaking corresponds to aninfinitely smooth winding-unwinding topological transition for the {\itclassical} statistical mechanics of a one-dimensional system: this describesthe the tangling/untangling of floppy polymers under a biasing torque. When theratio between torque and temperature exceeds a critical value the polymerundergoes tangled oscillations, with an extensive winding number. At lowertorque or higher temperature the winding number per unit length is zero.Approaching criticality, the correlation length of the order parameter---theextensive winding number---follows a Kosterlitz-Thouless type law. The model isdescribed by the Wilson line of a (0+1) $U(1)$ gauge theory, and applies to thetangling/untangling of floppy polymers and to the winding/diffusing kinetics indiffusion-convection-reactions.

Attractive Inverse Square Potential, U(1) Gauge, and Winding Transitions