In the 1980s Hubbard proposed a compelling question: when composing a polynomial known as the rabbit polynomial with a homeomorphism, the resulting map is equivalent to another (possibly different) polynomial. This question remained open for 25 years until Bartholdi–Nekrashevych solved in in 2006. In this talk, I highlight work with Mukundan and with Lanier that uses the combinatorial topological algorithm developed with Belk, Lanier, and Margalit to solve higher degree analogues of Hubbard's twisted rabbit problem. Our solutions reveal structure within unicritical degree-d polynomials in which the critical point is periodic and introduce accessible questions. This is a joint meeting of the dynamics seminar + the geometry seminar.