On $\mathbb{Z}_2$-Thurston norms and pseudo-horizontal surfaces in orientable Seifert $3$-manifolds

TL;DR

This paper introduces a method to compute the $ Z_2$-Thurston norm in orientable Seifert 3-manifolds using pseudo-horizontal surfaces, providing criteria, calculations, and algorithms for $ Z_2$-homology classes.

Contribution

It develops a general algorithm and criteria for calculating the $ Z_2$-Thurston norm via pseudo-horizontal surfaces in orientable Seifert manifolds, advancing understanding of their topology.

Findings

Established necessary and sufficient conditions for pseudo-horizontal surfaces.

Calculated non-orientable genera of these surfaces.

Provided an algorithm for computing the $ Z_2$-Thurston norm.

Abstract

We describe a general method to compute the $\mathbb{Z}_2$-Thurston norm for every $\mathbb{Z}_2$-homology class in an orientable Seifert manifold with orientable orbit surface. Our main tools are pseudo-horizontal surfaces. We give a necessary and sufficient criterion for the existence of pseudo-horizontal surfaces, calculate the non-orientable genera for such surfaces, and detect their $\mathbb{Z}_2$-homology classes. We then describe an algorithm to calculate the $\mathbb{Z}_2$-Thurston norm of each $\mathbb{Z}_2$-homology classes. We also present several interesting examples.