The complexity of Platonic and Archimedean polyhedral links

Abstract

A mathematical methodology for understanding the complexity of Platonic and Archimedean polyhedral links has been developed based on some topological invariants from knot theory. Knot invariants discussed here include crossing number, unknotting number, genus and braid index, which are considered significant in view of DNA nanotechnology. Our results demonstrate that the braid index provides the most structural information; hence, it can be used, among four knot invariants, as the most useful complexity measure. Using such an invariant, it indicates that the complexity of polyhedral links is directed by the number of their building blocks. The research introduces a simple but important concept in the theoretical characterization and analysis of DNA polyhedral catenanes.