Alternating knots do not admit cosmetic crossings

By examining the homology groups of a 4-manifold associated to an integral surgery on a knot $K$ in a rational homology 3-sphere $Y$ yielding a rational homology 3-sphere $Y^*$ with surgery dual knot $K^*$, we show that the subgroups generated by $[K]$ and $[K^*]$ in $H_1(Y)$ and $H_1(Y^*)$, respectively, have co-prime orders. We obtain an immediate corollary that, in conjunction with an argument of Lidman and Moore, proves the cosmetic crossing conjecture for knots whose branched double covers are Heegaard Floer L-spaces.