  The inverse square potential arises in a variety of different quantum
phenomena, yet notoriously it must be handled with care: it suffers from
pathologies rooted in the mathematical foundations of quantum mechanics. We
show that its recently studied conformality-breaking corresponds to an
infinitely smooth winding-unwinding topological transition for the {\it
classical} statistical mechanics of a one-dimensional system: this describes
the the tangling/untangling of floppy polymers under a biasing torque. When the
ratio between torque and temperature exceeds a critical value the polymer
undergoes tangled oscillations, with an extensive winding number. At lower
torque or higher temperature the winding number per unit length is zero.
Approaching criticality, the correlation length of the order parameter---the
extensive winding number---follows a Kosterlitz-Thouless type law. The model is
described by the Wilson line of a (0+1) $U(1)$ gauge theory, and applies to the
tangling/untangling of floppy polymers and to the winding/diffusing kinetics in
diffusion-convection-reactions.
