Conway spheres as ideal points of the character variety
It was proved by \textit{M. Culler} and \textit{P. B. Shalen} in [Ann. Math. (2) 117, 109--146 (1983; Zbl 0529.57005)] that essential surfaces in \(3\)-manifolds are associated to ideal points of the character variety. However not all essential surfaces can be detected this way. In this paper, the authors prove that a special family of essential surfaces, namely Conway spheres, correspond to an ideal point of the character variety. More precisely, if \(L\) is a hyperbolic link admitting a unique Conway sphere \(C\) up to isotopy and if \(C'\) is the planar surface which is the intersection of \(C\) with the exterior of the link, then \(C'\) corresponds to an ideal point of the character variety of the exterior of \(L\). This result and its generalization to hyperbolic links admitting more than one Conway sphere is proved by studying properties of the pieces obtained by cutting the link exterior along Conway spheres and their character varieties and by exhibiting irreducible characters in the intersection of certain varieties which do not have a limit in the character variety of the link exterior.
scientific article; zbMATH DE number 6087535
scientific article