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Register a new userThe surface complex of Seifert fibered spaces
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Authors
Jennifer Schultens
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We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex isomorphic to the curve complex of the base orbifold.
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Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.
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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