We examine the conjecture, due to Champanerkar, Kofman, and Purcell, 2015, that \vol[K] < 2 π log \det (K) for alternating hyperbolic links, where \vol[K] = \vol[S³\backslash K] is the hyperbolic volume and \det(K) is the determinant of K. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.

Supported by NSF Grants DMS-1105843 and DMS-1404754.