The Architecture And Topological Analysis Of Origami Links

After classifying and discussing about the majority new geometry structures synthesized by DNA origami strategy, we aimed to construct a new species of links that we defined as origami links (include 2D and 3D). Then we analyzed the component numbers of the origami links that is an invariant. In the end, a simple proof that any connected graph can be designed by DNA origami method was given in theory.First, the two-dimensional origami links have been constructed by our method "tangles covering". In order to facilitate the architecture of the links, tangles have been substituted by 4 degree vertices. Then a new term of links has been defined as "boundary". After that, the relation between the twist-number of tangle and the components of the links has been presented.Second, Platonic polyhedral links were constructed. Two methods were discussed:one was that nesting and assembling the known planar origami links into desired shapes; the other was that composed with a Hamilton circle travels all faces of polyhedron. The former one need to design edges and vertexes, respectively, but the later should be consider as a whole. "Cluster" and "Family" have been defined to classify Hamilton circles simultaneously.Finally, we got a real condition that a 2-regular graph had been established in a connected shape with random boundary, meaning that there was a Hamilton circle. Then the graph could extend to a Hamilton graph. At the end, we could get an origami link. It was proved that the shape could be synthesized by DNA origami method in theory.