A filtered mapping cone formula for cables of the knot meridian

TL;DR

This paper develops a filtered mapping cone formula to compute the knot Floer complex for cables of the knot meridian in rational surgeries, extending previous results and enabling new applications in knot concordance.

Contribution

It introduces a generalized filtered mapping cone formula for cables of the knot meridian in rational surgeries, broadening the scope of prior large surgery results.

Findings

Constructed a filtered mapping cone formula for $(n,1)$-cables in rational surgeries.

Showed existence of knots with arbitrary $\

varphi_{i,j}\

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 $\varphi_{i,j}$ values for any $i>j\geq 0$, where $\varphi_{i,j}$ are the concordance homomorphisms defined by Dai-Hom-Stoffregen-Truong.