The classification of toroidal Dehn surgeries on Montesinos knots (Q413400)

Let \(K\) be a hyperbolic knot in \(S^3\). By Thurston's Hyperbolic Dehn Surgery Theorem, all Dehn surgeries on \(K\) except finitely many produce hyperbolic manifolds; those that are not hyperbolic are called exceptional surgeries. By Perelman's proof of the Geometrization Conjecture, it follows that a surgery is exceptional if and only if the resulting manifold is reducible, toroidal or it is a small Seifert fibered manifold. A Montesinos knot is formed by taking the denominator of a partial sum of several rational tangles, this is an interesting family of knots. By work of \textit{Y-Q. Wu} [J. Differ. Geom. 43, No. 1, 171--197 (1996; Zbl 0851.57018)], it is known that there are no exceptional surgeries on Montesinos knots of length at least 4. Also, by the work of \textit{M. Brittenham} and \textit{Y-Q. Wu} [Commun. Anal. Geom. 9, No.1, 97--113 (2001; Zbl 0964.57013)], all exceptional surgeries on Montesinos knots of length two, i.e. 2-bridge knots, have been determined. So, what remains to be done is a classification of exceptional surgeries on Montesions knots of length 3. There is no reducible surgery in those knots, for they are strongly invertible [\textit{M. Eudave-Muñoz}, Trans. Am. Math. Soc. 330, No. 2, 463--501 (1992; Zbl 0778.57004)]. There are some Seifert fibered manifold surgeries, but those have not been completely determined. In the paper under review a classification is done of all toroidal surgeries on Montesinos knots of length 3. The proof is technical, it is based on Hatcher-Oertel's algorithm for finding boundary slopes on Montesinos knots [\textit{A. E. Hatcher} and \textit{U. Oertel}, Topology 28, No.4, 453-480 (1989; Zbl 0686.57006)].