Contractibility of the Kakimizu complex and symmetric Seifert surfaces

Abstract

Kakimizu complex of a knot is a flag simplicial complex whose vertices correspond to minimal genus Seifert surfaces and edges to disjoint pairs of such surfaces. We discuss a general setting in which one can define a similar complex. We prove that this complex is contractible, which was conjectured by Kakimizu. More generally, the fixed-point set (in the Kakimizu complex) for any subgroup of an appropriate mapping class group is contractible or empty. Moreover, we prove that this fixed-point set is non-empty for finite subgroups, which implies the existence of symmetric Seifert surfaces.

arXiv comment24 pages, 7 figures

Cited by in corpus (6)

• Realisation and dismantlability
• The Kakimizu complex of a connected sum of links
• Graphs with convex balls
• The coarse geometry of the Kakimizu complex
• The Kakimizu complex for genus one hyperbolic knots in the 3-sphere
• Non-split, alternating links bound unique Seifert surfaces in the 4-ball