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Ebeling and Ploog \cite{EbelingPloog} studied a duality of bimodularsingularities which is part of the Berglund--H$\ddot{\textnormal{u}}$bschmirror symmetry. Mase and Ueda \cite{MU} showed that this duality leads to apolytope mirror symmetry of families of $K3$ surfaces. We discuss in thisarticle how this symmetry extends to a symmetry between lattices.

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