A filtered mapping cone formula for cables of the knot meridian

Abstract

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 in the work of Dai–Hom–Stoffregen–Truong. This formula also leads to the construction of knots in integer homology spheres that bound PL surfaces with arbitrarily large genus in a homology ball in the work of Hom–Stoffregen–Zhou.