Inventiones mathematicae

, Volume 109, Issue 1, pp 473–494 | Cite as

Cusp structures of alternating links

  • I. R. Aitchison
  • E. Lumsden
  • J. H. Rubinstein


An alternating link ℒ Г is canonically associated with every finite, connected, planar graph Γ. The natural ideal polyhedral decomposition of the complement of ℒ Г is investigated. Natural singular geometric structures exist onS3−ℒ Г , with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection. For the class of ‘balanced graphs’, this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles. Examples of links containing immersed, closed π1-injective surfaces in their complements are given. These surfaces persist after ‘most’ surgeries on the link, the resulting closed 3-manifolds consequently being determined by their fundamental groups.


Planar Graph Geometric Structure Fundamental Group Equilateral Triangle Flat Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AR1] Aitchison, I.R., Rubinstein, J.H.: An introduction to polyhedral metrics of non-positive curvature on 3-manifolds. In: Geometry of Low-Dimensional Manifolds, vol. II: Symplectic Manifolds and Jones-Witten Theory pp. 127–161. Cambridge: Cambridge University Press, 1990Google Scholar
  2. [AR2] Aitchison, I.R., Rubinstein, J.H.: Combinatorial cubings, cusps, and the dodecahedral knots. In: Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. Topology90 (to appear)Google Scholar
  3. [AR3] Aitchison, I.R., Rubinstein, J.H.: Canonical surgery on alternating link diagrams, In: Proceedings of the International Conference on Knots, Osaka 1990 (to appear)Google Scholar
  4. [AR4] Aitchison, I.R., Rubinstein, J.H.: Polyhedral metrics of non-positive curvature on 3-manifolds with cusps. (In preparation)Google Scholar
  5. [BGS] Ballman, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature. Boston: Birkhäuser 1985Google Scholar
  6. [BH] Bleiler, S., Hodgson, C.: Spherical space forms and Dehn surgery. Proceedings of the International Conference on Knots, Osaka 1990 (to appear)Google Scholar
  7. [Co1] Coxeter, H.S.M.: Regular Polytopes. London: Methuen & Co. 1948Google Scholar
  8. [Co2] Coxeter, H.S.M.: Regular honeycombs in hyperbolic space. In: Proc. I.C.M., 1954. Amsterdam: North-Holland 1956Google Scholar
  9. [Gr] Gromov, M.: Hyperbolic manifolds groups and actions. Riemann surfaces and related topics. In: Kra, I., Maskit, B. (eds.) Stonybrook Conference Proceedings. (Ann. Math. Stud., vol. 97, pp. 183–214) Princeton: Princeton University Press 1981Google Scholar
  10. [GT] Gromov, M., Thurston, W.P.: Pinching constants for hyperbolic manifolds. Invent. Math.89, 1–12 (1987)Google Scholar
  11. [HKW] de la Harpe, P., Kervaire, M., Weber, C.: On the Jones polynomial. Enseign. Math.32, 271–335 (1986)Google Scholar
  12. [HS] Hass, J., Scott, P.: Homotopy equivalence and homeomorphism of 3-manifolds. (Preprint MSRI July 1989)Google Scholar
  13. [HRS] Hass, J., Rubinstein, H., Scott, P.: Covering spaces of 3-manifolds. Bull. Am. Math. Soc.16, 117–119 (1987)Google Scholar
  14. [Ha] Hatcher, A.: Hyperbolic structures of arithmetic type on some link complements. J. Lond. Math. Soc.27, 345–355 (1983)Google Scholar
  15. [Ho1] Hodgson, C.: Notes on the orbifold thorem. (In preparation)Google Scholar
  16. [Ho2] Hodgson, C.: Private communication. (Melbourne 1989)Google Scholar
  17. [La] Lawson, T.C.: Representing link complements by identified polyhedra. (Preprint)Google Scholar
  18. [Me1] Menasco, W.W.: Polyhedra representation of link complements. (Contemp Math., vol. 20, pp. 305–325) Providence, RI: Am. Math. Soc. 1983Google Scholar
  19. [Me2] Menasco, W.W.: Closed incompressible surfaces in alternating knot and link complements. Topology23, 37–44 (1984)Google Scholar
  20. [Re] Reid, A.W.: Totally geodesic surfaces in hyperbolic 3-manifolds. (Preprint); Proc Edinb. Math. Soc. (to appear)Google Scholar
  21. [Ro] Rolfsen, D.: Knots and Links. Berkeley: Publish or Perish 1976Google Scholar
  22. [Ta] Takahashi, M.: On the concrete construction of hyperbolic structures of 3-manifolds. (Preprint)Google Scholar
  23. [Th] Thurston, W.P.: The geometry and topology of 3-manifolds. Princeton University Lecture Notes 1978Google Scholar
  24. [Wa] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56–88 (1968)Google Scholar
  25. [We1] Weeks, J.R.: Hyperbolic structures on three-manifolds. PhD dissertation, Princeton 1985Google Scholar
  26. [We2] Weeks, J.R.: Programs for hyperbolic structures on three-manifolds. Macintosh II version 1990Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • I. R. Aitchison
    • 1
  • E. Lumsden
    • 1
  • J. H. Rubinstein
    • 1
  1. 1.Department of MathematicsUniversity of MelbourneParkvilleAustralia

Personalised recommendations