Exceptional surgeries on $(-2,p,p)$-pretzel knots

TL;DR

This paper classifies all exceptional surgeries on $(-2,p,p)$-pretzel knots with $p \,\geq\, 5$, showing they admit a unique toroidal surgery and no Seifert fibered surgeries, with detailed topological descriptions.

Contribution

It provides a complete characterization of exceptional surgeries on $(-2,p,p)$-pretzel knots, including the uniqueness of the toroidal surgery and the absence of Seifert fibered surgeries.

Findings

Unique toroidal surgery for these knots with a specific incompressible torus.

The other component after cutting is a twisted $I$-bundle over the Klein bottle.

No Seifert fibered surgeries exist for these pretzel knots.

Abstract

We give a complete description of exceptional surgeries on pretzel knots of type $(-2, p, p)$ with $p \ge 5$. It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a twisted $I$-bundle over the Klein bottle. We show that the other is homeomorphic to the one obtained by certain Dehn filling on the magic manifold. On the other hand, we show that all such pretzel knots admit no Seifert fibered surgeries.