A note on L-spaces which are double branched covers of non-quasi-alternating links

Greene has given an infinite family of non-quasi-alternating links Lm,n in which L2,3 is homologically thin and its double branched cover X2,3 is an L-space. In this note we show that the double branched cover Xm,n of Lm,n is an L-space for all n>m≥2 and Lm,n is the unique link whose double branched cover is Xm,n.