A hyperbolic approach to exp_3(S^1)

S. C. F. Rose

Published 2007-08-15Version 1

In this paper we investigate a new geometric method of studying exp_k(S^1), the set of all non-empty subsets of the circle of cardinality at most k. By considering the circle as the boundary of the hyperbolic plane we are able to use its group of isometries to determine explicitely the structure of its first few configuration spaces. We then study how these configuration spaces fit together in their union, exp_3(S^1), to reprove an old theorem of Bott as well as to offer a new proof (following that of E. Shchepin) of the fact that the embedding exp_1(S^1) into exp_3(S^1) is the trefoil knot.