The parameter space $S_p$ for monic centered cubic polynomial maps with a marked critical point of period p is a smooth affine algebraic curve whose genus increases rapidly with p. Each $S_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note with describe the topology of $S_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $S_p$.