End-essential spanning surfaces for links in thickened surfaces

Let $D$ be a cellular alternating link diagram on a closed orientable surface $\mathrm{\Sigma }$. 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 $\mathrm{\partial }$-parallel in $\mathrm{\Sigma }\times I$. 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.