Article

Inventiones mathematicae

, Volume 109, Issue 1, pp 473-494

First online:

Cusp structures of alternating links

  • I. R. AitchisonAffiliated withDepartment of Mathematics, University of Melbourne
  • , E. LumsdenAffiliated withDepartment of Mathematics, University of Melbourne
  • , J. H. RubinsteinAffiliated withDepartment of Mathematics, University of Melbourne

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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.