  We carry out ray tracing through five high resolution simulations of a galaxy
cluster to study how its ability to produce giant gravitationally lensed arcs
is influenced by the collision cross-section of its dark matter. In three cases
typical dark matter particles in the cluster core undergo between 1 and 100
collisions per Hubble time; two more explore the long (``collisionless'') and
short (``fluid'') mean free path limits. We study the size and shape
distributions of arcs and compute the cross-section for producing ``extreme''
arcs of various sizes. Even a few collisions per particle modify the core
structure enough to destroy the cluster's ability to produce long, thin arcs.
For larger collision frequencies the cluster must be scaled up to
unrealistically large masses before it regains the ability to produce giant
arcs. None of our models with self-interacting dark matter (except the
``fluid'' limit) is able to produce radial arcs; even the case with the
smallest scattering cross-section must be scaled to the upper limit of observed
cluster masses before it produces radial arcs. Apparently the elastic collision
cross-section of dark matter in clusters must be very small, below 0.1 cm^2
g^-1, to be compatible with the observed ability of clusters to produce both
radial arcs and giant arcs.
