End-essential spanning surfaces for links in thickened surfaces

TL;DR

This paper proves that checkerboard surfaces derived from certain alternating link diagrams on closed surfaces are topologically essential, supporting the study of virtual links and Turaev surfaces.

Contribution

It establishes the $ ext{π}_1$-essentiality of checkerboard surfaces for non-nugatory alternating diagrams on closed surfaces, advancing understanding of link surfaces in thickened surfaces.

Findings

Checkerboard surfaces are $ ext{π}_1$-essential.

No essential $ ext{∂}$-parallel closed curves in these surfaces.

Supports Tait's flyping conjecture for virtual links.

Abstract

Let $D$ be a cellular alternating link diagram on a closed orientable surface $\Sigma$. We prove that if $D$ has no removable nugatory crossings then each checkerboard surface from $D$ is $\pi_1$-essential and contains no essential closed curve that is $\partial$-parallel in $\Sigma\times I$. Our chief motivation comes from technical aspects of a companion paper, where we prove that Tait's flyping conjecture holds for alternating virtual links. We also describe possible applications via Turaev surfaces.