Knot reversal and rational concordance

TL;DR

This paper constructs an infinite family of knots that are not rationally concordant to their reverses, revealing new insights into the structure of the rational knot concordance group and the behavior of cable knots.

Contribution

It introduces an infinite family of knots with nontrivial reversal properties and demonstrates that the subgroup fixed under reversal has infinite rank within the rational concordance group.

Findings

Existence of knots not rationally concordant to their reverses

Infinite rank subgroup fixed under reversal in QC

Cabling operations preserve non-concordance to reverses

Abstract

We give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if R denotes the involution of the rational knot concordance group QC induced by string reversal and Fix(R) denotes the subgroup of knots fixed under R in QC, then QC/Fix(R) contains an infinite rank subgroup. As a corollary, we show that there exists a knot K such that for every pair of coprime integers p and q, the (p,q)-cable of K is not concordant to the reverse of the (p,q)-cable of K.