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 S^3, 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.