Covering spaces and the Kakimizu complex

Details Covering spaces and the Kakimizu complex Covering spaces and the Kakimizu complex

2007-07-26

arxiv.org/abs/0707.3926v3

Mathematics

Geometric Topology

16 pages, 10 figures

Scientific paper

We consider how the universal abelian cover of a knot exterior sheds light on the Kakimizu complex of the knot. First, we introduce the notion of covering spread for pairs of Seifert surfaces of a knot and prove that it equals the distance less 1 of the corresponding vertices in the Kakimizu complex. This provides an effective way of computing distances in the Kakimizu complex. Second, we prove that the Kakimizu complex is simply connected. Finally, we show that if the Kakimizu complex of a knot is at most 2-dimensional, then it is contractible.

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