Dehn surgery on knots of wrapping number $2$

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 $K_n\left(r_n\right)$ are toroidal for all but at most three $n$, or they are all atoroidal and nonhyperbolic. These will be used to classify exceptional surgeries on wrapped Montesinos knots in a solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus.