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Finite surgeries on three-tangle Pretzel knots. (English) Zbl 1166.57003

T. W. Mattman [J. Knot Theory Ramifications 11, No. 6, 891–902 (2002; Zbl 1023.57016)] almost finished the determination of Pretzel knots admitting non-trivial finite Dehn surgery. In fact, he could not exclude the \((-2,p,q)\)-Pretzel knot where \(p\) and \(q\) are odd and \(5\leq p\leq q\). The paper under review concludes that this family does not admit non-trivial finite surgery. Thus, the only non-trivial finite Dehn surgeries on hyperbolic Pretzel knots are 17, 18, 19-surgeries on the \((-2,3,7)\)-Pretzel knot, and 22, 23-surgeries on the \((-2,3,9)\)-Pretzel knot.
The argument is as follows. First, assume \(7\leq p\leq q\) or \(p=5\) and \(q\geq 11\). Then the 6-theorem by I. Agol [Geom. Topol. 4, 431–449 (2000; Zbl 0959.57009)] and M. Lackenby [Invent. Math. 140, No. 2, 243–282 (2000; Zbl 0947.57016)] eliminates almost all slopes. The length of a slope is calculated based on the construction of an ideal triangulation of the knot complement and a cusp cross-section. The remaining candidates are excluded by Culler-Shalen norm arguments and some group theoretic calculations.
Hence, three knots, \((-2,5,5)\), \((-2,5,7)\) and \((-2,5,9)\), remain. For the \((-2,5,5)\)-Pretzel knot, there are five boundary slopes by A. E. Hatcher and U. Oertel [Topology 28, No. 4, 453–480 (1989; Zbl 0686.57006)]. The detection of those boundary slopes except for one is shown by a similar argument to Y. Kabaya [A method to find ideal points from ideal triangulations, preprint]. Then the calculation of the Culler-Shalen norm implies that the trivial slope is the only finite surgery. The case for the \((-2,5,9)\)-Pretzel knot is similar. However, the case for the \((-2,5,7)\)-Pretzel knot needs extra information for the number of ideal points for some boundary slope. This is carried out by the method of T. Ohtsuki [J. Math. Soc. Japan 46, No. 1, 51–87 (1994; Zbl 0837.57006) and Topology Appl. 93, No. 2, 131–159 (1999; Zbl 0924.57004)].
The same result was independently obtained by K. Ichihara and I. D. Jong [Algebr. Geom. Topol. 9, No. 2, 731–742 (2009; Zbl 1165.57006)]. The argument is different, and they further determine all finite surgeries for Montesinos knots.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57M05 Fundamental group, presentations, free differential calculus

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References:

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