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 Topolog