Immersed surfaces and Seifert fibered surgery on Montesinos knots (Q2838072)

Exceptional Dehn surgeries have been classified for arborescent knots of length at least 4 in the present author's paper [Pac. J. Math. 252, No. 1, 219--243 (2011; Zbl 1233.57004)], and for 2-bridge knots [\textit{M. Brittenham} and \textit{Y-Q. Wu}, Commun. Anal. Geom. 9, No. 1, 97--113 (2001; Zbl 0964.57013)]. There is no reducible surgery on hyperbolic arborescent knots [the author, J. Differ. Geom. 43, No. 1, 171--197 (1996; Zbl 0851.57018)], and toroidal Dehn surgeries on length 3 Montesinos knots have also been determined [the author, Commun. Anal. Geom. 19, No. 2, 305--346 (2011; Zbl 1250.57018)].NEWLINENEWLINEIn the paper under review the author gives a classification of atoroidal Seifert fibered surgery on length 3 Montesinos knots \(K(p_1/q_1,p_2/q_2,p_3/q_3)\). As the author says, atoroidal Seifert fibered surgery is much more difficult to deal with than other types of exceptional surgeries. The major difficulty in dealing with it is that there are no embedded essential small surfaces (sphere, disk, annulus or torus) in such manifolds, and therefore one cannot use those traditional combinatorial methods on intersection graphs for such surgery problems. In this paper, however, the author uses immersed surfaces as a major tool. He studies the intersection of an immersed surface \(F\) with the tangle decomposition surfaces and the tangle spaces, and shows that no such surface could exist when the tangles are not too simple.NEWLINENEWLINEThe main theorem says: Let \(K= K(p_1/q_1,p_2/q_2, p_3/q_3)\) be a hyperbolic Montesinos knot of length 3, if \({1\over {q_1-1}} + {1\over {q_2-1}} + {1\over {q_3-1}}\leq 1\), then \(K\) admits no atoroidal Seifert fibered surgery. In particular he gives the following result, considering that two knots \(K,K'\) are equivalent if \(K\) is isotopic to \(K'\) or its mirror image. Suppose a Montesinos knot \(K\) of length 3 admits an atoroidal Seifert fibered surgery, then \(K\) is equivalent to one of the following knots:NEWLINENEWLINE (1) \(K(1/3,\pm 1/4, p_3/5)\) and \(p_3 \equiv \pm1\hskip 3pt mod\hskip 4pt 5\);NEWLINENEWLINE (2) \(K(1/3,\pm 1/3, p_3/q_3)\) and \(| \bar p_3|\leq 2\);NEWLINENEWLINE (3) \(K(1/2, 2/5, p_3/q_3)\) \(q_3 =5 \) or \(7\), and \(|\bar p_3|>1\);NEWLINENEWLINE (4) \(K(1/2,1/q_2, p_3/q_3), q_2\geq 5\) and \(| \bar p_3 | \leq 2\);NEWLINENEWLINE (5) \(K(1/2, 1/4, p_3/q_3)\) and \(| \bar p_3 | \leq 6\),NEWLINENEWLINE where \(\bar p_3\) denotes the mod \(q_3\) inverse of \(-p_3\) with minimal absolute value.NEWLINENEWLINE \textit{K. Ichihara} and \textit{In Dae Jong} [Commun. Anal. Geom. 18, No. 3, 579--600 (2010; Zbl 1234.57005)] showed that the only toroidal Seifert fibered surgery on Montesinos knots is the \(0\) surgery on the trefoil knot, hence the main theorem is still true with the word ``atoroidal'' deleted, and there is no reducible surgery on hyperbolic Montesinos knots [op. cit.], so the following result follows from the main theorem of the paper: Suppose \(K= K(p_1/q_1,p_2/q_2, p_3/q_3)\) is a hyperbolic Montesinos knot of length 3. If \({1\over {q_1-1}} + {1\over {q_2-1}} + {1\over {q_3-1}}\leq 1\) and a Dehn surgery \(K(r)\) is nonhyperbolic, then \(|p_i| = 1\), \(r\) is the pretzel slope, and \(K(r)\) is toroidal.NEWLINENEWLINEBy the results of this paper, plus some results recently announced by Ying-Qing Wu, J. Meier and K. Ichihara and H. Masai, there is now a complete classification of exceptional surgeries on length 3 Montesinos knots, and hence a complete classification of all exceptional surgeries on arborescent knots.