Supernovae of all types are known to be polarized, and many display complex line
polarization effects that evolve over time as the supernova evolves. Such behavior
reveals details of the clumpy nature of the ejecta, as well as illuminating the
characteristics of the circumstellar material lost by the star in its pre-supernova
evolution.  With the aid of diagnostic tools developed through numerical modeling,
polarized supernova spectra have the potential to reveal otherwise inaccessible
information about ejecta and surrounding environments.

To illustrate this process, I will present results from a grid of simulated polarized
line profiles of core-collapse SNe with circumstellar material, created using a
three-dimensional spectropolarimetric radiative transfer code. Taking into account
spectropolarimetric line effects helps break the degeneracy between line synthesis
models and quantify the asphericities often invoked to explain asymmetric line
profiles in total light spectra.  In addition, three-dimensional modeling can help
interpret the mysterious “q-u loop” phenomenon seen in the polarized line profiles of
a variety of recent SNe.  I will show simultaneous fits to observed line profiles in
total and polarized light, investigate the conditions under which q-u loops arise in
these models, and discuss the implications of these simulations for interpretation of
the polarized spectra of future supernovae.