Seifert fibered surgery and Rasmussen invariant (Q2867798)

Let \(K\) be a hyperbolic knot in the \(3\)-sphere \(S^3\). Then Thurston's hyperbolic Dehn surgery theorem says that all but finitely many Dehn surgeries on \(K\) produce hyperbolic \(3\)-manifolds. Furthermore, thanks to the positive solution of the Geometrization conjecture any exceptional (non-hyperbolic) surgery is a reducible surgery, a toroidal surgery or a Seifert fibered surgery. Thus it is interesting to determine all exceptional surgeries on hyperbolic knots.NEWLINENEWLINEIn [\textit{K. Ichihara} et al., Topology Appl. 159, No. 4, 1064--1073 (2012; Zbl 1244.57015)] the authors and \textit{Y. Kabaya} classify exceptional surgeries on pretzel knots \(P(-2, q, q)\) with \(q\geq 5\). In particular, they have shown that these pretzel knots do not admit Seifert fibered surgery. In the paper under review, the authors prove that pretzel knots \(P(p, q, q)\) with \(p, q\geq 2\) do not admit Seifert fibered surgery either. In the proof, they use a branched covering technique and the Montesinos trick [\textit{J. M. Montesinos}, Knots, Groups, 3-Manif.; Pap. dedic. Mem. R.H. Fox, 227--259 (1975; Zbl 0325.55004)], and apply a new criterion for a given knot to be a Montesinos knot using the Rasmussen invariant and the signature.NEWLINENEWLINERecently \textit{J. Meier} [Algebr. Geom. Topol. 14, No. 1, 439--487 (2014; Zbl 1290.57014)] has given a complete classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. For similar classifications for Montesinos knots, which include pretzel knots, see [\textit{J. Meier}, loc. cit.; \textit{Y.-Q. Wu}, ``Seifert fibered surgery on Montesinos knots'', \url{arXiv:1207.0154}; \textit{K. Ichihara} and \textit{H. Masai}, ``Exceptional surgeries on alternating knots'', \url{arXiv:1310.3472}].NEWLINENEWLINEFor the entire collection see [Zbl 1272.57002].