Concordance invariants of null-homologous knots in thickened surfaces

Abstract

Using the Gordon-Litherland pairing, one can define invariants (signature, nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. In this paper, we study the concordance properties of these invariants. For example, if $K \subset \Sigma \times I$ is ${\mathbb Z}/2$ null-homologous and slice, we show that its signatures vanish and its determinants are perfect squares. These statements are derived from a cobordism result for closed unoriented surfaces in certain 4-manifolds. The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. They take values in ${\mathbb Z}/8 \cup \{\infty\}$ and depend on a choice of spanning surface. We present two equivalent methods to defining and computing them, and we prove a chromatic duality result relating the two. We study their concordance properties, and we show how to interpret them as Arf invariants for null-homologous links. The Brown invariants and knot signatures are shown to be invariant under concordance of spanning surfaces.