DNA cages can be joined together to make larger 3D nanostructures on which molecular electronic circuits and tiny containers are built for drug delivery. The mathematical models for these promising nanomaterials play important roles in clarifying their assembly mechanism and understanding their structures. In this study, we propose a mathematical and computer method to construct permissible topological structures with double-helical edges for a triangular bipyramid and pentangular bipyramid. Furthermore, we remove the same topological links, without eliminating the nonrepeated ones for a triangular bipyramid and pentangular bipyramid. By analyzing characteristics of these unique links, some self-assembly and statistic rules are discussed. This study may obtain some new insights into the DNA assembly from the viewpoint of mathematics, promoting the comprehending and design efficiency of DNA polyhedra with required topological structures.

中文翻译：

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三角双锥和矩形双锥的双螺旋结构及分析。

可以将DNA笼子连接在一起以形成更大的3D纳米结构，并在其上构建分子电子电路和微型容器来进行药物输送。这些有前途的纳米材料的数学模型在阐明其组装机理和理解其结构方面起着重要作用。在这项研究中，我们提出了一种数学和计算机方法来构造三角形双锥体和矩形双锥体的具有双螺旋边缘的允许拓扑结构。此外，我们删除了相同的拓扑链接，但没有消除三角形双锥和矩形双锥的非重复链接。通过分析这些独特链接的特征，讨论了一些自组装和统计规则。这项研究可能会从数学的角度获得一些有关DNA组装的新见解，