Large alternating Montesinos knots do not admit purely cosmetic surgeries

Abstract

It is conjectured that, on a non-trivial knot in the 3-sphere, no pair of Dehn surgeries along distinct slopes are purely cosmetic, that is, none of them yield 3-manifolds those are orientation-preservingly homeomorphic. In this paper, we show that alternating knots having reduced alternating diagram with the twist number at least 7, Montesinos knots of length at least 5, and alternating Montesinos knots of length at least 4 do not admit purely cosmetic surgeries. As a corollary, we see that large alternating Montesinos knots have no purely cosmetic surgeries.