A note on orientation-reversing distance one surgeries on non-null-homologous knots

We show that there are no distance one surgeries on non-null-homologous knots in M M that yield − M -M ( M M with opposite orientation) if M M is a 3-manifold obtained by a Dehn surgery on a knot K K in S 3 S³ , such that the order of its first homology is divisible by 9 9 but is not divisible by 27 27 . As an application, we show several knots, including the ( 2 , 9 ) (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the ( 2 , k ) (2,k) torus knot ( k > 1 ) (k>1) has a chirally cosmetic banding if and only if k = 5 k=5 .