Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques — adapting tools used tostudy mapping class groups — to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.