The non-orientable genus (a.k.a crosscap number) of a knot is the smallest genus over all non-orientable surfaces spanned by the knot. In this talk, I’ll describe joint work with Christine Lee, in which we obtain two-sided linear bound of the crosscap number of alternating link in terms of the Jones link polynomial. The bounds are often exact and they allow us to compute the crosscap numbers of infinite families of alternating knots as well as the crosscap number of 283 knots with up to twelve crossings that were previously unknown. Time permitting, we will also discuss generalizations to families of non-alternating links. The proofs of the results use techniques from angled polyhedra decomposition of 3-manifolds, normal surface theory and the geometry of augmented links. The first talk, by Jessica Purcell, will in particular provide some background.