  We study a model of densely packed self-avoiding loops on the annulus,
related to the Temperley Lieb algebra with an extra idempotent boundary
generator. Four different weights are given to the loops, depending on their
homotopy class and whether they touch the outer rim of the annulus. When the
weight of a contractible bulk loop x = q + 1/q satisfies -2 < x <= 2, this
model is conformally invariant for any real weight of the remaining three
parameters. We classify the conformal boundary conditions and give exact
expressions for the corresponding boundary scaling dimensions. The amplitudes
with which the sectors with any prescribed number and types of non contractible
loops appear in the full partition function Z are computed rigorously. Based on
this, we write a number of identities involving Z which hold true for any
finite size. When the weight of a contractible boundary loop y takes certain
discrete values, y_r = [r+1]_q / [r]_q with r integer, other identities
involving the standard characters K_{r,s} of the Virasoro algebra are
established. The connection with Dirichlet and Neumann boundary conditions in
the O(n) model is discussed in detail, and new scaling dimensions are derived.
When q is a root of unity and y = y_r, exact connections with the A_m type RSOS
model are made. These involve precise relations between the spectra of the loop
and RSOS model transfer matrices, valid in finite size. Finally, the results
where y=y_r are related to the theory of Temperley Lieb cabling.
