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 $m{\mathcal{G}}^{\mathbb{Z}}$ and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection $\lambda :v\mathcal{C}\to m{\mathcal{G}}^{\mathbb{Z}}$, where $v\mathcal{C}$ 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 $m{\mathcal{G}}^{\mathbb{Z}}$ contains a copy of ${\mathbb{Z}}^{\infty }\oplus (\mathbb{Z}/2{)}^{\infty }\oplus (\mathbb{Z}/4{)}^{\infty }.$

arXiv comment34 pages, 8 figures. Revisions made to section 2