# Cusp structures of alternating links

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

## 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 on*S*^{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 3-manifolds consequently being determined by their fundamental groups.