Concordance invariants of null-homologous knots in thickened surfaces

TL;DR

This paper investigates the properties of certain invariants like signature, nullity, determinant, and Brown invariants for null-homologous links in thickened surfaces, revealing their behavior under concordance and their relation to cobordism and Arf invariants.

Contribution

It introduces new methods to compute Brown invariants, explores their concordance invariance, and relates them to classical invariants and cobordism results.

Findings

Signatures vanish for slice null-homologous links.

Determinants of slice links are perfect squares.

Brown invariants can be interpreted as Arf invariants.

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.