A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic and Archimedean solids by using our method of the ‘three-cross-curve and double-twist-line covering’. There are five classes of polyhedral links that can be explored: the tetrahedral and truncated tetrahedral links; the hexahedral and truncated hexahedral links; the dodecahedral and truncated dodecahedral links; the truncated octahedral and icosahedral links. Our results show that the tetrahedral and truncated tetrahedral links have T symmetry; the hexahedral and truncated hexahedral links, as well as the truncated octahedral links, O symmetry; the dodecahedral and truncated dodecahedral links, as well as the truncated icosahedral links, I symmetry, respectively. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA and protein catenanes.