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.

Hans U. Boden

Homayun Karimi

Communications in Analysis and Geometry

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The journal "Communications in Analysis and Geometry" publishes high-quality research papers related to classical analysis, partial differential equations, algebraic geometry, differential geometry, and topology.

《Communications in Analysis and Geometry》期刊发表与经典分析、偏微分方程、代数几何、微分几何和拓扑相关的高质量研究论文。

Category	Quartile
数学	4
数学, 数学	4

Concordance invariants of null-homologous knots in thickened surfaces

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