Contractibility of the Kakimizu complex and symmetric Seifert surfaces

Przytycki, Piotr; Schultens, Jennifer

doi:10.1090/S0002-9947-2011-05465-6

Contractibility of the Kakimizu complex and symmetric Seifert surfaces
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by Piotr Przytycki and Jennifer Schultens

Trans. Amer. Math. Soc. 364 (2012), 1489-1508

DOI: https://doi.org/10.1090/S0002-9947-2011-05465-6

Published electronically: October 27, 2011

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Abstract:

The 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.

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