Cusp structures of alternating links

Abstract

An alternating link ℒG{cyrillic} is canonically associated with every finite, connected, planar graph Γ. The natural ideal polyhedral decomposition of the complement of ℒG{cyrillic} is investigated. Natural singular geometric structures exist on S3-ℒG{cyrillic}, 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, th..