Classification & applications of the algebraic concordance of almost classical knots

As is well known, J. Levine classified the algebraic concordance group $\mathscr{G}^{\mathbb{F}}$ of knots in the $3$-sphere, where $\mathbb{F}$ is a field of characteristic 0. In this talk, we will define two generalizations of the algebraic concordance group for homologically trivial knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. The generalizations are called the coupled algebraic concordance group and the uncoupled algebraic concordance group. These can be realized as concordance classes of Seifert surfaces. For the uncoupled algebraic concordance group $\mathscr{VG}^{\mathbb{F}}$, we prove that $\mathscr{VG}^{\mathbb{F}} \cong \mathscr{I}(\mathbb{F})\oplus \mathscr{G}^{\mathbb{F}}$, where $\mathscr{F}$ is the fundamental ideal of the Witt ring over $\mathbb{F}$. For $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$, we also define an Arf invariant. Examples will be given in the several cases, with applications to virtual knots.