Tokyo Journal of Mathematics

Homology Cylinders and Sutured Manifolds for Homologically Fibered Knots

Hiroshi GODA and Takuya SAKASAI

Full-text: Open access


Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered knots.

Article information

Tokyo J. Math., Volume 36, Number 1 (2013), 85-111.

First available in Project Euclid: 22 July 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M05: Fundamental group, presentations, free differential calculus 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}


GODA, Hiroshi; SAKASAI, Takuya. Homology Cylinders and Sutured Manifolds for Homologically Fibered Knots. Tokyo J. Math. 36 (2013), no. 1, 85--111. doi:10.3836/tjm/1374497513.

Export citation


  • G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co., Berlin, 2003.
  • T. Cochran, K. Orr and P. Teichner, Knot concordance, Whitney towers and $L^2$-signatures, Ann. of Math. 157 (2003), 433–519.
  • R. Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959), 258–275.
  • R. Crowell and H. Trotter, A class of pretzel knots, Duke Math. J. 30 (1963), 373–377.
  • D. B. A. Epstein, Finite presentations of groups and $3$-manifolds, Quart. J. Math. Oxford 12 (1961), 205–212.
  • S. Friedl, Reidemeister torsion, the Thurston norm and Harvey's invariants, Pacific J. Math. 230 (2007), 271–296.
  • S. Friedl and T. Kim, The Thurston norm, fibered manifolds and twisted Alexander polynomials, Topology 45 (2006), 929–953.
  • D. Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), 445–503.
  • D. Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv. 61 (1986), 519–555.
  • D. Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), 479–536.
  • S. Garoufalidis and J. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in mathematics and theorical physics, Proc. Sympos. Pure Math. 73 (2005), 173–205.
  • H. Goda, Heegaard splitting for sutured manifolds and Murasugi sum, Osaka J. Math. 29 (1992), 21–40.
  • H. Goda, On handle number of Seifert surfaces in $S\sp 3$, Osaka J. Math. 30 (1993), 63–80.
  • H. Goda, Circle valued Morse theory for knots and links, Floer homology, gauge theory, and low-dimensional topology, 71–99, Clay Math. Proc. 5, Amer. Math. Soc., Providence, RI, 2006.
  • H. Goda, Some estimates of the Morse-Novikov numbers for knots and links, Carter, J. Scott et al. (ed.), Intelligence of low dimensional topology 2006, World Scientific, Series on Knots and Everything 40, 35–42, 2007.
  • H. Goda and M. Ishiwata, A classification of Seifert surfaces for some pretzel links, Kobe J. Math. 23 (2006), 11–28.
  • H. Goda and T. Sakasai, Abelian quotients of monoids of homology cylinders, Geometriae Dedicata 151 (2011), 387–396.
  • H. Goda and T. Sakasai, Factorization formulas and computations of higher-order Alexander invariants for homologically fibered knots, J. Knot Theory Ramifications 20 (2011), 1355–1380.
  • M. Goussarov, Finite type invariants and n-equivalence of 3-manifolds, C. R. Math. Acad. Sci. Paris 329 (1999), 517–522.
  • N. Habegger, Milnor, Johnson, and tree level perturbative invariants, preprint.
  • K. Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1–83.
  • J. Harlander and J. A. Jensen, On the homotopy type of CW-complexes with aspherical fundamental group, Topology Appl. 153 (2006), 3000–3006.
  • N. V. Ivanov, Mapping class groups, Handbook of geometric topology, 523–633, North-Holland, Amsterdam, 2002.
  • A. Kawauchi, On the integral homology of infinite cyclic coverings of links, Kobe J. Math. 4 (1987), 31–41.
  • A. Kawauchi (ed.), A survey of knot theory, Birkhauser Verlag, Basel, 1996.
  • P. Kirk, C. Livingston and Z. Wang, The Gassner representation for string links, Commun. Contemp. Math. 3 (2001), 87–136.
  • J. Y. Le Dimet, Enlacements d'intervalles et torsion de Whitehead, Bull. Soc. Math. France 129 (2001), 215–235.
  • J. Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001), 243–270.
  • W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer-Verlag, New York, 1997.
  • C. T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 153–171.
  • J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. 76 (1962), 137–147.
  • J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.
  • S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699–726.
  • K. Murasugi, On the genus of the alternating knot, I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248.
  • K. Murasugi, On a certain subgroup of the group of an alternating link, Amer. J. Math. 85 (1963), 544–550.
  • D. Passman, The Algebraic Structure of Group Rings, John Wiley and Sons, 1977.
  • T. Sakasai, The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces, Algebr. Geom. Topol. 8 (2008), 803–848.
  • R. Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974), 302–332.
  • V. Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2001).
  • K. Veber, A. Pazhitnov and L. Rudolf, The Morse-Novikov number for knots and links, (Russian) Algebra i Analiz 13 (2001), 105–118; translation in St. Petersburg Math. J. 13 (2002), 417–426.
  • F. Waldhausen, Whitehead groups of generalized free products, Lecture Notes in Mathematics 342, Springer-Verlag, Berlin (1973), 155–179.