## Algebraic & Geometric Topology

### The coarse geometry of the Kakimizu complex

#### Abstract

We show that the Kakimizu complex of minimal genus Seifert surfaces for a knot in the $3$–sphere is quasi-isometric to a Euclidean integer lattice $ℤn$ for some $n≥0$.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 5 (2014), 2549-2560.

Dates
Revised: 31 January 2014
Accepted: 7 February 2014
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715995

Digital Object Identifier
doi:10.2140/agt.2014.14.2549

Mathematical Reviews number (MathSciNet)
MR3276840

Zentralblatt MATH identifier
1302.57022

#### Citation

Johnson, Jesse; Pelayo, Roberto; Wilson, Robin. The coarse geometry of the Kakimizu complex. Algebr. Geom. Topol. 14 (2014), no. 5, 2549--2560. doi:10.2140/agt.2014.14.2549. https://projecteuclid.org/euclid.agt/1513715995

#### References

• J E Banks, On links with locally infinite Kakimizu complexes, Algebr. Geom. Topol. 11 (2011) 1445–1454
• J E Banks, The Kakimizu complex of a connected sum of links, Trans. Amer. Math. Soc. 365 (2013) 6017–6036
• R Budney, JSJ-decompositions of knot and link complements in $S\sp 3$, Enseign. Math. 52 (2006) 319–359
• P de la Harpe, Topics in geometric group theory, University of Chicago Press, Chicago, IL (2000)
• A Hatcher, Notes on basic $3$–manifold topology (2007) Available at \setbox0\makeatletter\@url http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf {\unhbox0
• O Kakimizu, Finding disjoint incompressible spanning surfaces for a link, Hiroshima Math. J. 22 (1992) 225–236
• O Kakimizu, Classification of the incompressible spanning surfaces for prime knots of 10 or less crossings, Hiroshima Math. J. 35 (2005) 47–92
• R C Pelayo, Diameter bounds on the complex of minimal genus seifert surfaces for hyperbolic knots, PhD thesis, California Institute of Technology (2007) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/1039265228 {\unhbox0
• P Przytycki, J Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces, Trans. Amer. Math. Soc. 364 (2012) 1489–1508
• M Sakuma, Minimal genus Seifert surfaces for special arborescent links, Osaka J. Math. 31 (1994) 861–905
• M Sakuma, K J Shackleton, On the distance between two Seifert surfaces of a knot, Osaka J. Math. 46 (2009) 203–221
• M Scharlemann, A Thompson, Finding disjoint Seifert surfaces, Bull. London Math. Soc. 20 (1988) 61–64
• R T Wilson, Knots with infinitely many incompressible Seifert surfaces, J. Knot Theory Ramifications 17 (2008) 537–551