End-essential spanning surfaces for links in thickened surfaces

Abstract

Let $D$ be a cellular alternating link diagram on a closed orientable surface $\hat{I}\pounds$. We prove that if $D$ has no removable nugatory crossings then each checkerboard surface from $D$ is $\ddot{I}{€}_1$-essential and contains no essential closed curve that is $\partial$-parallel in $\hat{I}\pounds \times I$. 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.

15 pages, 7 figures, comments welcome. Minor changes from the previous version