  The O(n) spin model in two dimensions may equivalently be formulated as a
loop model, and then mapped to a height model which is conjectured to flow
under the renormalization group to a conformal field theory (CFT). At the
critical point, the order n terms in the partition function and correlation
functions describe single self-avoiding loops. We investigate the ensemble of
these self-avoiding loops using twist operators, which count loops which wind
non-trivially around them with a factor -1. These turn out to have level two
null states and hence their correlators satisfy a set of partial differential
equations. We show that partly-connected parts of the four point function count
the expected number of loops which separate one pair of points from the other
pair, and find an explicit expression for this. We argue that the differential
equation satisfied by these expectation values should have an interpretation in
terms of a stochastic(Schramm)-Loewner evolution (SLE_kappa) process with
kappa=6. The two point function in a simply connected domain satisfies a
closely related set of equations. We solve these and hence calculate the
expected number of single loops which separate both points from the boundary.
