Algebraic concordance order of almost classical knots

TL;DR

This paper introduces a new algebraic framework for almost classical knots, classifies their concordance group, and reveals the existence of nontrivial finite-order elements not linked to classical invariants.

Contribution

It defines the virtual algebraic concordance group for almost classical knots and provides an algebraic classification, extending classical results to a broader virtual setting.

Findings

Embedding of classical algebraic concordance group into the virtual group

Existence of nontrivial finite-order elements outside classical concordance

Generalization of the Arf invariant for Z/2Z coefficients

Abstract

Torsion in the concordance group $\mathscr{C}$ of knots in $S^3$ can be studied with the algebraic concordance group $\mathscr{G}^{\mathbb{F}}$. Here $\mathbb{F}$ is a field of characteristic $\chi(\mathbb{F}) \ne 2$. The group $\mathscr{G}^{\mathbb{F}}$ was defined by J. Levine, who also obtained an algebraic classification when $\mathbb{F}=\mathbb{Q}$. While the concordance group $\mathscr{C}$ is abelian, it embeds into the non-abelian virtual knot concordance group $\mathscr{VC}$. It is unknown if $\mathscr{VC}$ admits non-classical finite torsion. Here we define the virtual algebraic concordance group $\mathscr{VG}^{\mathbb{F}}$ for almost classical knots . This is an analogue of $\mathscr{G}^{\mathbb{F}}$ for homologically trivial knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. The main result is an algebraic classification of $\mathscr{VG}^{\mathbb{F}}$. A consequence of the classification is that $\mathscr{G}^{\mathbb{Q}}$ embeds into $\mathscr{VG}^{\mathbb{Q}}$ and $\mathscr{VG}^{\mathbb{Q}}$ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$, we give a generalization of the Arf invariant.