On links with locally infinite {K}akimizu complexes

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.
Comment: 9 pages, 5 figures; v2 minor has minor changes incorporating referee's comments. To appear in Algebraic & Geometric Topology