Surgeries between lens spaces of type $L(n,1)$ and the Heegaard Floer $d$-invariant

We establish a $d$-invariant surgery formula for $L$-space knots that provides an effective tool for studying surgeries between lens spaces. Using this formula, we classify distance one surgeries between lens spaces of the form $L(n,1)$. This classification has direct applications to band surgeries between torus links $T(2,n)$, with connections to DNA topology. In particular, we show that chirally cosmetic banding of torus links can possibly occur only when $n=1,5,9$ or $10$.