  The main result in this paper is that the space of all smooth links in
Euclidean 3-space isotopic to the trivial link of n components has the same
homotopy type as its finite-dimensional subspace consisting of configurations
of n unlinked Euclidean circles (the "rings" in the title). There is also an
analogous result for spaces of arcs in upper half-space, with circles replaced
by semicircles (the "wickets" in the title). A key part of the proofs is a
procedure for greatly reducing the complexity of tangled configurations of
rings and wickets. This leads to simple methods for computing presentations for
the fundamental groups of these spaces of rings and wickets as well as various
interesting subspaces. The wicket spaces are also shown to be K(G,1)'s.
