Dehn surgery on knots of wrapping number 2

TL;DR

This paper investigates the conditions under which hyperbolic knots in a solid torus admit at most one exceptional, toroidal surgery, and explores implications for classifying surgeries on wrapped Montesinos knots.

Contribution

It establishes a bound on the number of exceptional surgeries for certain hyperbolic knots in a solid torus and applies this to classify surgeries on wrapped Montesinos knots.

Findings

At most one exceptional toroidal surgery for specified hyperbolic knots.

Infinite families of knots with related surgery properties.

Classification of exceptional surgeries on wrapped Montesinos knots.

Abstract

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K \cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embedding $V$ in $S^3$ gives infinitely many knots $K_n$ with a slope $r_n$ corresponding to a slope $r$ of $K$ in $V$. If $r$ surgery on $K$ in $V$ is toroidal then either all but at most three $K_n(r_n)$ are toroidal, or they are all reducible or small Seifert fibered with two common singular fiber indices. These will be used to classify exceptional surgeries on wrapped Montesinos knots in solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus.