Geometriae Dedicata

, Volume 168, Issue 1, pp 207–220 | Cite as

On the fibration of augmented link complements

Original paper


We study the fibration of augmented link complements. Given the diagram of an augmented link we associate a spanning surface and a graph. We then show that this surface is a fiber for the link complement if and only if the associated graph is a tree. We further show that fibration is preserved under Dehn filling on certain components of these links. This last result is then used to prove that within a very large class of links, called locally alternating augmented links, every link is fibered.


Link complements Fibered manifolds 

Mathematics Subject Classification

57M05 57M25 



I am very grateful to Alan Reid for his guidance during my graduate program. I am also thankful to Cameron Gordon for helpful conversations and João Nogueira and Jessica Purcell for their comments on an early draft of this work. Finally I would like to thank the referee for his careful reading of this paper and his many comments which helped improve it.


  1. 1.
    Adams, C.: Augmented Alternating Link Complements are Hyperbolic, Low-Dimensional Topology and Kleinian Groups (Coventry-Durham, 1984), London Mathematical Society Lecture Notes Series, vol. 112, pp. 115–130. Cambridge University Press, Cambridge (1986)Google Scholar
  2. 2.
    DeBlois, J., Chesebro, E., Wilton, H.: Some virtually special hyperbolic 3-manifold groups. Commentarii Mathematici Helvetici 87(3), 727–787 (2012)MATHMathSciNetGoogle Scholar
  3. 3.
    Futer, D.: Fiber detection for state surfaces (preprint)Google Scholar
  4. 4.
    Futer, D., Kalfagianni, E., Purcell, J.: Dehn filling, volume and the Jones polynomial. J. Differ. Geom. 78(3), 429–464 (2008)MATHMathSciNetGoogle Scholar
  5. 5.
    Futer, D., Kalfagianni, E., Purcell, J.: Guts of surfaces and the colored Jones polynomial. In: Monograph to appear in Lecture Notes in Mathematics, vol. 2069. Springer, BerlinGoogle Scholar
  6. 6.
    Futer, D., Purcell, J.: Links with no exceptional surgeries. Comment. Math. Helvetici 8(3), 629–664 (2007)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gabai, D.: Detecting fibred links in \(S^3\). Comment. Math. Helvetici 61, 519–555 (1986)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Gabai, D.: The Murasugi sum is a natural geometric operation II. Comtemporary Math. 20, 131–143 (1983)MATHMathSciNetGoogle Scholar
  9. 9.
    Goodman, S., Tavares, G.: Pretzel-fibered links. Bol. Soc. Bras. Mat., vol 15, no 1 e 2, pp. 85–96 (1984)Google Scholar
  10. 10.
    Harer, J.: How to construct all fibered knots and links. Topology 21, 263–280 (1982)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lackenby, M.: The volume of hyperbolic alternating link complements. With an appendix by Ian Agol and Dylan Thurston. Proc. Lond. Math. Soc. 88, 204–224 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Melvin, P.M., Morton, H.R.: Fibred knots of genus 2 formed by plumbing Hopf bands. J. Lond. Math. Soc. 34(2), 159–168 (1986)Google Scholar
  13. 13.
    Murasugi, K.: On a certain subgroup of the group of an alternating link. Am. J. Math. 85, 544–550 (1963)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Purcell, J.: An introduction to fully augmented links. Interact. Hyperbolic Geom. Quantum Topol. Number Theory Contemp. Math. 541, 205–220 (2011)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Purcell, J.: Volumes of highly twisted knots and links. Algebraic Geom. Topol. 7, 93–108 (2007)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Rolfsen, D.: Knots and Links. AMS Chelsea Publishing, Providence (2003)Google Scholar
  17. 17.
    Stallings, J.: On Fibering of Certain 3-manifolds Topology of 3-manifolds and Related Topics. Prentice-Hall, Englewood Cliffs (1962)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversidade Federal do CearáFortaleza, CearáBrazil

Personalised recommendations