Splittings of knot groups

Abstract

Let be a knot of genus . If is fibered, then it is well known that the knot group splits only over a free group of rank . We show that if is not fibered, then splits over non-free groups of arbitrarily large rank. Furthermore, if is not fibered, then splits over every free group of rank at least . However, cannot split over a group of rank less than . The last statement is proved using recent results of Agol, Przytycki-Wise and Wise.