We introduce a new DNA tile self-assembly model: the Surface Flexible Tile Assembly Model (SFTAM), where 2D tiles are placed on host 3D surfaces made of axis-parallel unit cubes glued together by their faces, called polycubes. The bonds are flexible, so that the assembly can bind on the edges of the polycube. We are interested in the study of SFTAM self-assemblies on 3D surfaces which are not always embeddable in the Euclidean plane, in order to compare their different behaviors and to compute the topological properties of the host surfaces. We focus on a family of polycubes called cuboids. Order-0 cuboids are polycubes that have six rectangular faces, and order-1 cuboids are made from two order-0 cuboids by substracting one from the other. Thus, order-1 cuboids can be of genus 0 or of genus 1 (then they contain a tunnel). We are interested in the genus of these structures, and we present a SFTAM tile assembly system that determines the genus of a given order-1 cuboid. The SFTAM tile assembly system which we design, contains a specific set $Y$ of tile types with the following properties. If the assembly is made on a host order-1 cuboid $C$ of genus 0, no tile of $Y$ appears in any producible assembly, but if $C$ has genus 1, every terminal assembly contains at least one tile of $Y$. Thus, we are able to distinguish the host surfaces according to their genus, by the tiles used in the assembly. This system is specific to order-1 cuboids but the techniques we use should be generalizable to other families of shapes.

翻译：我们提出了一种新的DNA瓦片自组装模型：表面柔性瓦片组装模型（SFTAM），其中二维瓦片被放置在由轴平行单位立方体通过面粘合形成的宿主三维曲面上，这种结构称为多立方体。键具有柔性，因此组装体能够结合在多立方体的边缘上。我们关注SFTAM在并非总能嵌入欧几里得平面的三维曲面上的自组装行为，旨在比较其不同表现并计算宿主曲面的拓扑性质。重点研究一类称为长方体的多立方体。零阶长方体是具有六个矩形面的多立方体，一阶长方体则由两个零阶长方体通过相减操作构成。因此，一阶长方体可能具有亏格0或亏格1（此时包含一个隧道）。我们关注这些结构的亏格，并提出一个SFTAM瓦片组装系统，用于确定给定一阶长方体的亏格。我们设计的SFTAM瓦片组装系统包含一个特定的瓦片类型集合$Y$，具有以下性质：若组装发生在亏格0的宿主一阶长方体$C$上，任何可生成的组装体中均不出现$Y$中的瓦片；但若$C$的亏格为1，每个终端组装体至少包含一个$Y$中的瓦片。由此，我们能够通过组装中使用的瓦片来区分宿主曲面的亏格。该系统针对一阶长方体设计，但所采用的技术应可推广至其他形状族。