  We study the loop representation of the quantum theory for 2+1 dimensional
general relativity on a manifold, $M = {\cal T}^2 \times {\cal R}$, where
${\cal T}^2$ is the torus, and compare it with the connection representation
for this system. In particular, we look at the loop transform in the part of
the phase space where the holonomies are boosts and study its kernel. This
kernel is dense in the connection representation and the transform is not
continuous with respect to the natural topologies, even in its domain of
definition. Nonetheless, loop representations isomorphic to the connection
representation corresponding to this part of the phase space can still be
constructed if due care is taken. We present this construction but note that
certain ambiguities remain; in particular, functions of loops cannot be
uniquely associated with functions of connections.
