##
**Cusp structures of alternating links.**
*(English)*
Zbl 0810.57010

Given a finite, connected graph \(\Gamma\) in a 2-sphere in \(S^ 3\) there is naturally associated an alternating link \({\mathcal L}_ \Gamma\) whose projection to the 2-sphere meets each complementary region of \(\Gamma\) in its inscribed polygon. There is an associated decomposition of the complement \(S^ 3 - {\mathcal L}_ \Gamma\) into ideal polyhedra as introduced by Thurston in his notes on “The Geometry and Topology of 3- manifolds”. This paper is a study of these constructions as tools in the understanding of the 3-manifolds \(S^ 3 - {\mathcal L}_ \Gamma\) and their Dehn fillings relative to questions such as the existence of incompressible surfaces, the existence of (possibly singular) metrics of negative curvature, determination by their fundamental group, and so on. Numerous examples of these phenomena are provided.

Reviewer: J.Hempel (Houston)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

### Keywords:

finite, connected graph in a 2-sphere; link complements; alternating link; ideal polyhedra; Dehn fillings; existence of incompressible surfaces; metrics of negative curvature; fundamental group
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\textit{I. R. Aitchison} et al., Invent. Math. 109, No. 3, 473--494 (1992; Zbl 0810.57010)

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