Algebraic & Geometric Topology

On sutured Floer homology and the equivalence of Seifert surfaces

Matthew Hedden, András Juhász, and Sucharit Sarkar

Full-text: Open access


The goal of this paper is twofold. First, given a Seifert surface R in the 3–sphere, we show how to construct a Heegaard diagram for the sutured manifold S3(R) complementary to R, which in turn enables us to compute the sutured Floer homology of S3(R) combinatorially. Secondly, we outline how the sutured Floer homology of S3(R), together with the Seifert form of R, can be used to decide whether two minimal genus Seifert surfaces of a given knot are isotopic in S3. We illustrate our techniques by showing that the knot 83 has two minimal genus Seifert surfaces up to isotopy. Furthermore, for any n1 we exhibit a knot Kn that has at least n nonisotopic free minimal genus Seifert surfaces.

Article information

Algebr. Geom. Topol., Volume 13, Number 1 (2013), 505-548.

Received: 9 March 2011
Accepted: 7 October 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

Heegaard diagram Seifert surface Floer homology


Hedden, Matthew; Juhász, András; Sarkar, Sucharit. On sutured Floer homology and the equivalence of Seifert surfaces. Algebr. Geom. Topol. 13 (2013), no. 1, 505--548. doi:10.2140/agt.2013.13.505.

Export citation


  • W R Alford, Complements of minimal spanning surfaces of knots are not unique, Ann. of Math. 91 (1970) 419–424
  • I Altman, Sutured Floer homology distinguishes between Seifert surfaces, Topology Appl. 159 (2012) 3143–3155
  • J R Eisner, Knots with infinitely many minimal spanning surfaces, Trans. Amer. Math. Soc. 229 (1977) 329–349
  • S Friedl, A Juhász, J Rasmussen, The decategorification of sutured Floer homology, J. Topol. 4 (2011) 431–478
  • D Gabai, Foliations and genera of links, PhD thesis, Princeton University (1980) Available at \setbox0\makeatletter\@url {\unhbox0
  • D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
  • D Gabai, The Murasugi sum is a natural geometric operation, from: “Low-dimensional topology”, (S J Lomonaco, Jr, editor), Contemp. Math. 20, Amer. Math. Soc. (1983) 131–143
  • D Gabai, Detecting fibred links in $S\sp 3$, Comment. Math. Helv. 61 (1986) 519–555
  • D Gabai, Foliations and the topology of $3$–manifolds. II, J. Differential Geom. 26 (1987) 461–478
  • P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151–1169
  • A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429–1457
  • A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299–350
  • A Juhász, The sutured Floer homology polytope, Geom. Topol. 14 (2010) 1303–1354
  • 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
  • T Kobayashi, Uniqueness of minimal genus Seifert surfaces for links, Topology Appl. 33 (1989) 265–279
  • R Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006) 955–1097
  • H C Lyon, Simple knots without unique minimal surfaces, Proc. Amer. Math. Soc. 43 (1974) 449–454
  • C Manolescu, P Ozsváth, S Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. 169 (2009) 633–660
  • Y Ni, Sutured Heegaard diagrams for knots, Algebr. Geom. Topol. 6 (2006) 513–537
  • Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577–608
  • Y Ni, Erratum: Knot Floer homology detects fibred knots, Invent. Math. 177 (2009) 235–238
  • P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225–254
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003) Available at \setbox0\makeatletter\@url {\unhbox0
  • S Sarkar, J Wang, An algorithm for computing some Heegaard Floer homologies, Ann. of Math. 171 (2010) 1213–1236
  • W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130
  • H F Trotter, Some knots spanned by more than one knotted surface of mininal genus, from: “Knots, groups, and $3$–manifolds”, (L P Neuwirth, editor), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 51–52
  • V Turaev, Torsion invariants of ${\rm Spin}\sp c$–structures on $3$–manifolds, Math. Res. Lett. 4 (1997) 679–695