Octahedral decomposition of alternating links in thickened surfaces

The octahedral decomposition of classical link complements has been considered and utilised by Weeks, Rubinstein, Sakuma etc. It is even more natural to consider the octahedral decomposition of virtual link complements. In this paper, we explore and generalise the octahedral decomposition used in \emph{snappy} to links in thickened surface. Using the decomposition, we prove nonpositive curvature of the complement and essential-ness of edges in the decomposition.