How to determine the number and folding pathway of ssDNA strands assembling DNA polyhedrons are the fundamental problems in DNA nanoassembly. In this paper, the related mathematical problems are presented by introducing antiparallel oriented links (AO links) to describe the topological structures of DNA polyhedrons, that is, how to give all AO link diagrams based on any 2-connected plane graph and their link components. We demonstrate that each AO link diagram must be isotopy equivalent to a special AO link diagram with even tangle edges by defining ``Node Move'' operations on vertex nodes. By further giving the relationship between AO link diagrams and edge-weighted plane graphs, two algorithms are established to calculate all AO link diagrams and their link components. Based on these two algorithms, a software "AO link" has been developed in Fortran language, giving all AO link diagrams and their link components only by inputting a planar graph. Also, the other three versions of the software are established to eliminate isotopy classes of AO link diagrams. This work not only reveals some important properties of AO link diagrams but also provides an important tool for related chemical and mathematical problems.