Algebraic concordance order of almost classical knots

Abstract

Torsion in the concordance group [Formula: see text] of knots in [Formula: see text] can be studied with the algebraic concordance group [Formula: see text]. Here, [Formula: see text] is a field of characteristic [Formula: see text]. The group [Formula: see text] was defined by Levine, who also obtained an algebraic classification when [Formula: see text]. While the concordance group [Formula: see text] is abelian, it embeds into the non-abelian virtual knot concordance group [Formula: see text]. It is unknown if [Formula: see text] admits non-classical finite torsion. Here, we define the virtual algebraic concordance group [Formula: see text] for Seifert surfaces of almost classical knots. This is an analogue of [Formula: see text] for homologically trivial knots in thickened surfaces [Formula: see text], where [Formula: see text] is closed and oriented. The main result is an algebraic classification of [Formula: see text]. A consequence of the classification is that [Formula: see text] embeds into [Formula: see text] and [Formula: see text] contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For [Formula: see text], we give a generalization of the Arf invariant.