Dehn surgery on knots of wrapping number 2

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.