A mirror duality for families of $K3$ surfaces associated to bimodular singularities

Ebeling and Ploog \cite{EbelingPloog} studied a duality of bimodular singularities which is part of the Berglund–H$\ddot{\text{u}}$bsch mirror symmetry. Mase and Ueda \cite{MU} showed that this duality leads to a polytope mirror symmetry of families of $K3$ surfaces. We discuss in this article how this symmetry extends to a symmetry between lattices.