  In this paper, we examine the behavior of modes of the Kerr geometry when the
mode's frequency is purely imaginary. We demonstrate that quasinormal modes
must be polynomial in nature if their frequency is purely imaginary, and
present a method for computing such modes. The nature of these modes, however,
is not always easy to determine. Some of the polynomial modes we compute are
quasinormal modes. However, some are simultaneously quasinormal modes and total
transmission modes, while others fail to satisfy the requisite boundary
conditions for either. This analysis is, in part, an extension of the results
known for Schwarzschild black holes, but clarifies misconceptions for the
behavior of modes when the black hole has angular momentum. We also show that
the algebraically special modes of Kerr with m=0 have an additional branch of
solutions not seen before in the literature. All of these results are in
precise agreement with new numerical solutions for sequences of gravitational
quasinormal modes of Kerr. However, we show that some prior numerical and
analytic results concerning the existence of quasinormal modes of Kerr with
purely imaginary frequencies were incorrect.
