Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [Ban11] J. E. Banks, On links with locally infinite Kakimizu complexes (2011), available at arXiv:1010.3831.
  • [CO09] V. Chepoi and D. Osajda, Dismantlability of weakly systolic complexes and applications (2009), available at arXiv:0910.5444.
  • [Edm84] A. L. Edmonds, Least area Seifert surfaces and periodic knots, Topology Appl. 18 (1984), no. 2-3, 109-113. MR 769284 (86c:57005)
  • [HS97] M. Hirasawa and M. Sakuma, Minimal genus Seifert surfaces for alternating links, KNOTS '96 (Tokyo), World Sci. Publ., River Edge, NJ, 1997, pp. 383-394. MR 1664976 (2000e:57009)
  • [Kak92] O. Kakimizu, Finding disjoint incompressible spanning surfaces for a link, Hiroshima Math. J. 22 (1992), no. 2, 225-236. MR 1177053 (93k:57013)
  • [Kak05] O. Kakimizu, Classification of the incompressible spanning surfaces for prime knots of $ 10$ or less crossings, Hiroshima Math. J. 35 (2005), no. 1, 47-92. MR 2131376 (2005m:57011)
  • [Lüc05] W. Lück, Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 269-322. MR 2195456 (2006m:55036)
  • [Oer88] U. Oertel, Sums of incompressible surfaces, Proc. Amer. Math. Soc. 102 (1988), no. 3, 711-719. MR 929008 (89c:57013)
  • [Pel07] R. C. Pelayo, Diameter bounds on the complex of minimal genus Seifert surfaces for hyperbolic knots (2007), thesis at California Institute of Technology, available at
  • [Pol00] N. Polat, On infinite bridged graphs and strongly dismantlable graphs, Discrete Math. 211 (2000), no. 1-3, 153-166. MR 1735348 (2000k:05232)
  • [Sak94] M. Sakuma, Minimal genus Seifert surfaces for special arborescent links, Osaka J. Math. 31 (1994), no. 4, 861-905. MR 1315011 (96b:57011)
  • [SS09] M. Sakuma and K. J. Shackleton, On the distance between two Seifert surfaces of a knot, Osaka J. Math. 46 (2009), no. 1, 203-221. MR 2531146 (2010f:57016)
  • [Sch10] J. Schultens, The Kakimizu complex is simply connected (2010), preprint.
  • [ST88] M. Scharlemann and A. Thompson, Finding disjoint Seifert surfaces, Bull. London Math. Soc. 20 (1988), no. 1, 61-64. MR 916076 (89a:57007)
  • [Thu80] W. P. Thurston, The geometry and topology of three-manifolds (1980), available at

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M25

Retrieve articles in all journals with MSC (2010): 57M25

Additional Information

Piotr Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland

Jennifer Schultens
Affiliation: Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, California 95616

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.

American Mathematical Society