, Volume 109, Issue 1, pp 473494
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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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 π_{1}injective surfaces in their complements are given. These surfaces persist after ‘most’ surgeries on the link, the resulting closed 3manifolds consequently being determined by their fundamental groups.
 Title
 Cusp structures of alternating links
 Journal

Inventiones mathematicae
Volume 109, Issue 1 , pp 473494
 Cover Date
 199212
 DOI
 10.1007/BF01232034
 Print ISSN
 00209910
 Online ISSN
 14321297
 Publisher
 SpringerVerlag
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 Authors

 I. R. Aitchison ^{(1)}
 E. Lumsden ^{(1)}
 J. H. Rubinstein ^{(1)}
 Author Affiliations

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