The Kakimizu Complex for a knot is what you get by taking the definition of the curve complex for a surface and replacing loops in the surface with minimal genus Seifert surfaces for . It consists of a vertex for each isotopy class of minimal genus Seifert surface for with edges connecting any two vertices with disjoint representatives, and simplices spanning larger collections of pairwise disjoint surfaces. This complex turns out to be contractible  and for atoroidal knots, it’s finite . But something I found quite surprising is an example by Jessica Banks  whose Kakimizu complex is locally infinite. This knot is relatively simple and you can see a picture of it below the fold.
The knot is a satellite over the trefoil with the property that inside the solid torus defined by the trefoil, is homologically trivial. (Correction: In Bank’s minimal example, there is one fewer full twists in the band between the parallel trefoils.) That means that it has a Seifert surface contained in the solid torus (shown on the right). But it also has a minimal genus Seifert surface that cuts through the solid torus (shown on the left). In the picture, the colors indicate the two “sides” of the Seifert surface.
If you take the correct Haken sum of the surface on the left with a copy of the incompressible torus, you get a new minimal genus Seifert surface
that is disjoint from the two surfaces shown here. If you Haken sum the resulting surface with a second copy of the incompressible torus, the new surface will intersect the original surface on the left non-trivially, but will be disjoint from the surface on the right.
If you continue taking Haken sums, all the resulting surfaces will be disjoint from the original surface shown on the right. This follows from the fact that the right surface is disjoint from the incompressible torus. Banks shows that the Haken summed surfaces are isotopically distinct, so in the Kakimizu complex the vertex corresponding to the surface on the right has infinitely many edges coming out of it.
The portion of the Kakimizu complex spanned by these surfaces is isomorphic to a cone over the line with edges between consecutive integers. (Correction: After a brief e-mail discussion with Jessica Banks, I think this statement is incorrect. It seems that the complex for this knot is a cone over infinitely many discrete points, but I’m still worried I’m missing something.)