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Contractibility of the Kakimizu complex and symmetric Seifert surfaces


Authors: Piotr Przytycki and Jennifer Schultens
Journal: Trans. Amer. Math. Soc. 364 (2012), 1489-1508
MSC (2010): Primary 57M25
Published electronically: October 27, 2011
MathSciNet review: 2869183
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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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Additional Information

Piotr Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland
Email: pprzytyc@mimuw.edu.pl

Jennifer Schultens
Affiliation: Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, California 95616
Email: jcs@math.ucdavis.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05465-6
Received by editor(s): May 7, 2010
Received by editor(s) in revised form: June 2, 2010, and September 13, 2010
Published electronically: October 27, 2011
Additional Notes: The first author was partially supported by MNiSW grant N201 012 32/0718, MNiSW grant N N201 541738 and the Foundation for Polish Science.
The second author was partially supported by an NSF grant DMS-0905798.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.