Conformal "inside-out" transformations of torus knots

These knots are stereographic projections of knots on a flat torus in the 3-sphere. The torus can be turned inside out by rigid rotation around a (1,1) great circle on the torus. These rotations are after stereographic projection conformal transformations in R^3 which turn a (p,q)-knot into a (q,p)-knot. The corresponding inside-out transformation of the torus in R^3 passes through infinity. Transforming the knots in R^3 can be achieved by keeping them away from the point on the torus which passes through infinity.