Finite subset spaces of the circle and a theorem of Bott

By considering the circle as the boundary of the hyperbolic plane we are able to describe the first three unordered configuration spaces of the circle by considering them as particular quotients of the group of isometries of the hyperbolic plane. After determining how these join together and calculating their fundamental group, we describe their union exp_3(S1) as a simply connected Seifert-Fibred space, hence S3. Moreover, a slight variation of this method reveals that the inclusion of S1 into this space is in fact the trefoil knot.