  Strong lensing is a powerful probe of the distribution of matter in the cores
of clusters of galaxies. Recent studies suggest that the cold dark matter model
predicts cores that are denser than those observed in galaxies, groups and
clusters. One possible resolution of the discrepancy is that the dark matter
has strong interactions (SIDM), which leads to lower central densities. A
generalized form of the Navarro, Frenk and White profile (Zhao profile) may be
used to describe these halos. In this paper we examine gravitational lensing
statistics for this class of model. The optical depth to multiple imaging is a
very sensitive function of the profile parameters in the range of interest for
SIDM halos around clusters of galaxies. Less concentrated profiles, which
result from larger self-interaction cross-sections, can produce many fewer
lensed pairs. Lensing statistics provide a powerful test for SIDM. More
realistic and observationally oriented calculations remain to be done, however
larger self-interaction cross-sections may well be ruled out by the very
existence of strong lenses on galaxy cluster scales. The inclusion of centrally
dominant cluster galaxies should boost the cross-section to multiple imaging.
However our preliminary calculations suggest that the additional multiple
imaging rate is small with respect to the differences in multiple imaging rate
for different halo profiles. In future statistical studies, it will be
important to properly account for the scatter among halo profiles since the
optical depth to multiple imaging is dominated by the most concentrated members
of a cluster population.
