Skip to main content
SpringerLink
Log in

Cusp structures of alternating links

  • Published:
Inventiones mathematicae Aims and scope

Summary

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 onS 3−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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [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, 1990

    Google Scholar 

  • [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)

  • [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)

  • [AR4] Aitchison, I.R., Rubinstein, J.H.: Polyhedral metrics of non-positive curvature on 3-manifolds with cusps. (In preparation)

  • [BGS] Ballman, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature. Boston: Birkhäuser 1985

    Google Scholar 

  • [BH] Bleiler, S., Hodgson, C.: Spherical space forms and Dehn surgery. Proceedings of the International Conference on Knots, Osaka 1990 (to appear)

  • [Co1] Coxeter, H.S.M.: Regular Polytopes. London: Methuen & Co. 1948

    Google Scholar 

  • [Co2] Coxeter, H.S.M.: Regular honeycombs in hyperbolic space. In: Proc. I.C.M., 1954. Amsterdam: North-Holland 1956

    Google Scholar 

  • [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 1981

    Google Scholar 

  • [GT] Gromov, M., Thurston, W.P.: Pinching constants for hyperbolic manifolds. Invent. Math.89, 1–12 (1987)

    Google Scholar 

  • [HKW] de la Harpe, P., Kervaire, M., Weber, C.: On the Jones polynomial. Enseign. Math.32, 271–335 (1986)

    Google Scholar 

  • [HS] Hass, J., Scott, P.: Homotopy equivalence and homeomorphism of 3-manifolds. (Preprint MSRI July 1989)

  • [HRS] Hass, J., Rubinstein, H., Scott, P.: Covering spaces of 3-manifolds. Bull. Am. Math. Soc.16, 117–119 (1987)

    Google Scholar 

  • [Ha] Hatcher, A.: Hyperbolic structures of arithmetic type on some link complements. J. Lond. Math. Soc.27, 345–355 (1983)

    Google Scholar 

  • [Ho1] Hodgson, C.: Notes on the orbifold thorem. (In preparation)

  • [Ho2] Hodgson, C.: Private communication. (Melbourne 1989)

  • [La] Lawson, T.C.: Representing link complements by identified polyhedra. (Preprint)

  • [Me1] Menasco, W.W.: Polyhedra representation of link complements. (Contemp Math., vol. 20, pp. 305–325) Providence, RI: Am. Math. Soc. 1983

    Google Scholar 

  • [Me2] Menasco, W.W.: Closed incompressible surfaces in alternating knot and link complements. Topology23, 37–44 (1984)

    Google Scholar 

  • [Re] Reid, A.W.: Totally geodesic surfaces in hyperbolic 3-manifolds. (Preprint); Proc Edinb. Math. Soc. (to appear)

  • [Ro] Rolfsen, D.: Knots and Links. Berkeley: Publish or Perish 1976

    Google Scholar 

  • [Ta] Takahashi, M.: On the concrete construction of hyperbolic structures of 3-manifolds. (Preprint)

  • [Th] Thurston, W.P.: The geometry and topology of 3-manifolds. Princeton University Lecture Notes 1978

  • [Wa] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56–88 (1968)

    Google Scholar 

  • [We1] Weeks, J.R.: Hyperbolic structures on three-manifolds. PhD dissertation, Princeton 1985

  • [We2] Weeks, J.R.: Programs for hyperbolic structures on three-manifolds. Macintosh II version 1990

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Melbourne, 3052, Parkville, Victoria, Australia

    I. R. Aitchison, E. Lumsden & J. H. Rubinstein

Authors
  1. I. R. Aitchison
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Lumsden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. H. Rubinstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Oblatum 20-V-1991

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aitchison, I.R., Lumsden, E. & Rubinstein, J.H. Cusp structures of alternating links. Invent Math 109, 473–494 (1992). https://doi.org/10.1007/BF01232034

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01232034

Keywords

Search

Navigation