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 (see the Clifford Torus
for more details).Transforming the knots in R^3 can be achieved by keeping
them away from the point on the torus which passes through infinity.