Suppose is a hyperbolic knot in a solid torus intersecting a meridian disk twice. We will show that if is not the Whitehead knot and the frontier of a regular neighborhood of is incompressible in the knot exterior, then admits at most one exceptional surgery, which must be toroidal. Embedding in gives infinitely many knots with a slope corresponding to a slope of in . If surgery on in is toroidal then either are toroidal for all but at most three , 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.
"Dehn surgery on knots of wrapping number $2$." Algebr. Geom. Topol. 13 (1) 479 - 503, 2013. https://doi.org/10.2140/agt.2013.13.479