Knot reversal and rational concordance

Abstract

We give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if τ$\tau$ denotes the involution of the rational knot concordance group scriptCdouble-struckQ$\mathcal {C}_\mathbb {Q}$ induced by reversal and Fixfalse(τfalse)$\mbox{Fix}(\tau )$ denotes the subgroup of knots fixed under τ$\tau$ in scriptCdouble-struckQ$\mathcal {C}_\mathbb {Q}$, then CQ/Fix(τ)$\mathcal {C}_\mathbb {Q}/\mbox{Fix}(\tau )$ contains an infinite rank subgroup. As a corollary, we show that there exists a knot K$K$ such that for every pair of coprime integers p$p$ and q$q$, the false(p,qfalse)$(p,q)$‐cable of K$K$ is not concordant to the reverse of the false(p,qfalse)$(p,q)$‐cable of K$K$.