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

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

Published electronically:
October 27, 2011

MathSciNet review:
2869183

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Abstract | References | Similar Articles | Additional Information

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:
https://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.