In knot theory, a knot is a closed simple curve embedded in the 3D space and a link is a finite disjoint collection of knots [1]. With the development of DNA nanotechnology, these mathematical objects have been realized on the field of DNA polyhedral assembly [2,3]. Some covalently closed polyhedral links were assembled such that each oligonucleotide walking along exactly one face, such as DNA tetrahedra [4,5], DNA cube [6], DNA Triangular bipyramid [7], DNA Octahedra [8] and so on. Also, some polyhedral knots were formed by folding a long DNA single strand, such as DNA tetrahedron [9], DNA triangular prism [10], DNA pyramid [11], DNA Pentagonal pyramid [12] and so on. Hence for DNA polyhedron with double helical edges, some questions arise naturally that how many ssDNA strands they can be assembled from and what topological structures these strands may form. This paper will try to resolve these problems on DNA octahedrons.

Octahedra, as one of five Plato polyhedrons, have triangles as faces similar to tetrahedron and triangular bipyramid. However, octahedra has larger space capacity and more complex structure. These geometrical features increase the difficulty of structural synthesis of DNA octahedra [8,13,14]. Until 2004, a long DNA strand with a sequence designed specially was folded successfully into a 3D structure having octahedral shape with the help of five short strands [13]. However, this DNA octahedron has no special topological structure and does not contain any knots or links. After that, Knudsen and co-workers reported the first covalently closed DNA octahedral link assemble from eight different oligonucleotides [8]. Each oligonucleotide contains three 18-base-length subsequences that complement each other to form 12 double helix edges. This DNA octahedra is endowed with important application prospects due to its thermal and chemical stability [15]. Also, the related MD simulation has been reported to understand molecular mechanisms of topological structure assembly [[16], [17], [18]]. In fact, the link structure of this DNA octahedra was determined uniquely by the number of ssDNA strands from mathematical viewpoint. Hence to assemble new DNA octahedrons with double helix edges, we need to characterize the new topological structures firstly by changing the number of ssDNA strands.

Polyhedral links, as the interlinked and interlocked architectures in polyhedral shape, have served as mathematical models for descripting the structural properties of DNA polyhedra [[19], [20], [21], [22]]. In recent work, some polyhedral knots have been constructed based on Platonic polyhedra and truncated polyhedra [[23], [24], [25]]. More generally, all possible topological structures of DNA tetrahedrons, DNA trigonal prisms and DNA trigonal bipyramids have been characterized by establishing the related polyhedral link models [[26], [27], [28], [29], [30], [31]]. Among these works, the algorithm and Python program designed for DNA trigonal pyramid [29] provides an available approach to describing the topological structures of DNA octahedrons. However, how to determine the corresponding topological structures by the number of DNA strands is an unsolved important question.

In this paper, oriented octahedral links are established as the structural models of DNA octahedrons with double helix edges. Firstly, 36 orientations are determined by considering the orientation of each tangle edge successively. Then according to each orientation, 1566 types of oriented octahedral links are generated by choosing all matched tangle types of each edge from four basic building blocks. In this process, the repeated orientations or links produced by the symmetry operations of octahedra are excluded. At last, the component of each octahedral link can be calculated as a serial of vertex arcs by establishing a new algorithm. Here a program “Octa-links” is developed in the Python language, divided into three parts to achieve the results above. Our work provides a complete list of topological structures for new DNA octahedrons assembled by adjusting the number of ssDNA strands, as well as an available approach for the similar problems with other DNA polyhedrons.