Algebraic concordance order of almost classical knots

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