Inventiones mathematicae

, Volume 109, Issue 1, pp 473–494

Cusp structures of alternating links


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

DOI: 10.1007/BF01232034

Cite this article as:
Aitchison, I.R., Lumsden, E. & Rubinstein, J.H. Invent Math (1992) 109: 473. doi:10.1007/BF01232034


An alternating link ℒГ is canonically associated with every finite, connected, planar graph Γ. The natural ideal polyhedral decomposition of the complement of ℒГ is investigated. Natural singular geometric structures exist onS3−ℒГ, 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.

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© Springer-Verlag 1992