Obstructing two-torsion in the rational knot concordance group

TL;DR

This paper investigates the existence of 2-torsion elements in the rational knot concordance group, providing obstructions to their finite order status and constructing an infinite rank subgroup of such knots.

Contribution

It introduces an obstruction based on the localized von Neumann rho-invariant to distinguish knots of algebraic order 2 from those of finite order in the rational concordance group, and constructs an infinite subgroup.

Findings

Obstruction prevents certain algebraic 2-torsion knots from having finite order in the rational concordance group.

Constructs an infinite rank subgroup generated by knots with algebraic order 2.

Higher-dimensional cases show the algebraic and geometric rational concordance orders coincide.

Abstract

It is well known that there are many 2-torsion elements in the classical knot concordance group. On the other hand, it is not known if there is any torsion element in the rational knot concordance group $\mathcal{C}_\mathbb{Q}$. Cha defined the algebraic rational concordance group $\mathcal{AC}_\mathbb{Q}$, an analogue of the classical algebraic concordance group, and showed that $\mathcal{AC}_\mathbb{Q}\cong\mathbb{Z}^\infty\oplus\mathbb{Z}_2^\infty\oplus\mathbb{Z}_4^\infty$. The knots that represent 2-torsions in $\mathcal{AC}_\mathbb{Q}$ potentially have order $2$ in $\mathcal{C}_\mathbb{Q}$. In this paper, we provide an obstruction for knots of order $2$ in $\mathcal{AC}_\mathbb{Q}$ from being of finite order in $\mathcal{C}_\mathbb{Q}$. Moreover, we give a family consisting of such knots that generates an infinite rank subgroup of $\mathcal{C}_\mathbb{Q}$. We also note that Cha proved that in higher dimensions, the algebraic rational concordance order is the same as the rational knot concordance order. Our obstruction is based on the localized von Neumann $\rho$-invariant.