Small Seifert fibered surgery on hyperbolic pretzel knots

Abstract

The study of exceptional surgery on hyperbolic knots has been well developed over the last quarter century. One particularly well studied problem is that of exceptional surgery on arborescent knots, which include Montesinos knots and pretzel knots. Thanks to the positive solution to the geometrization conjecture by Perelman [38; 39; 40], any exceptional surgery is either reducible, toroidal, or a small Seifert fibered space. Exceptional surgeries on hyperbolic arborescent knots of length 4 or greater have been classified by Wu [52], as have exceptional surgeries on hyperbolic 2–bridge knots; see Brittenham and Wu [7]. It has been shown that no hyperbolic arborescent knot admits a reducible surgery (see Wu [48]), and toroidal surgeries on hyperbolic arborescent knots of length three are completely classified; see Wu [51].