Mock Seifert matrices and unoriented algebraic concordance

Abstract

A mock Seifert matrix is an integral square matrix representing the Gordon-Litherland form of a pair $(K,F)$, where $K$ is a knot in a thickened surface and $F$ is an unoriented spanning surface for $K$. Using these matrices, we introduce a new notion of unoriented algebraic concordance, as well as a new group denoted $\mathcal{m} \mathcal{G}^{\mathbb Z}$ and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection $\lambda \colon \mathcal{v} \mathcal{C} \to \mathcal{m} \mathcal{G}^{\mathbb Z}$, where $\mathcal{v} \mathcal{C} $ denotes the virtual knot concordance group. Mock Seifert matrices can also be used to define new invariants, such as the mock Alexander polynomial and mock Levine-Tristram signatures. These invariants are applied to questions about virtual knot concordance, crosscap numbers, and Seifert genus for knots in thickened surfaces. For example, we show that $\mathcal{m} \mathcal{G}^{\mathbb Z}$ contains a copy of ${\mathbb Z}^\infty \oplus ({\mathbb Z}/2)^\infty \oplus({\mathbb Z}/4)^\infty.$