A filtered mapping cone formula for cables of the knot meridian

We construct a filtered mapping cone formula that computes the knot Floer complex of the $(n,1)$–cable of the knot meridian in any rational surgery, generalizing Truong’s result about the $(n,1)$–cable of the knot meridian in large surgery and Hedden-Levine’s filtered mapping cone formula. As an application, we show that there exist knots in integer homology spheres with arbitrary ${\phi }_{i,j}$ values for any $i>j\ge 0$, where ${\phi }_{i,j}$ are the concordance homomorphisms defined by Dai-Hom-Stoffregen-Truong.