In this paper, the novel topology of Platonic polyhedral links is discussed on the basis of the graph theory and topological principles. This interesting problem of the dual polyhedral links has been solved by using our method of the ‘sphere-surface-movement’. There are three classes of dual polyhedral links which can be explored: the tetrahedral link is self-dual, the hexahedral and octahedral link, as well as the dodecahedral and icosahedral link are dual to each other. Our results show that the duality of self-dual tetrahedral link is ‘trivial’, and the duality of hexahedral and octahedral link as well as dodecahedral and icosahedral link are ‘nontrivial’. This study provides further insight into the molecular design and theoretical characterization of the new polyhedral links.