Napoleon in isolation (Q2723511)

This paper deals with isolation of cusps in finite volume hyperbolic 3-manifolds. Given such a manifold, if we assume that it is non-compact, then its ends are cusps, by the thin-thick decomposition. Isolation occurs when any hyperbolic Dehn filling in some of the cusps leaves invariant the geometric structure of some other cusps. This was an unattended phenomenon when \textit{W. D. Neumann} and \textit{A. W. Reid} found the first examples of manifolds with isolated cusps [Math. Ann. 295, No. 2, 223-237 (1993; Zbl 0813.57013)]. NEWLINENEWLINENEWLINEThe author gives a new construction of hyperbolic manifolds with isolated cusps. Namely he constructs a hyperbolic knot with six cusps that has two sets of cusps in Brunnian isolation (three cusps are in Brunnian isolation when any surgery in one of them leaves the geometry of the other two invariant, but a surgery in two of them can change the structure of the third). His construction is based in Napoleon's theorem in elementary geometry, that describes how certain operations on plane polygons always produce regular polygons. This is why he calls his example Napoleon manifold.