Which polyominoes can be folded into a cube, using only creases along edges of the square lattice underlying the polyomino, with fold angles of $\pm 90^\circ$ and $\pm 180^\circ$, and allowing faces of the cube to be covered multiple times? Prior results studied tree-shaped polyominoes and polyominoes with holes and gave partial classifications for these cases. We show that there is an algorithm deciding whether a given polyomino can be folded into a cube. This algorithm essentially amounts to trying all possible ways of mapping faces of the polyomino to faces of the cube, but (perhaps surprisingly) checking whether such a mapping corresponds to a valid folding is equivalent to the unlink recognition problem from topology. We also give further results on classes of polyominoes which can or cannot be folded into cubes. Our results include (1) a full characterisation of all tree-shaped polyominoes that can be folded into the cube (2) that any rectangular polyomino which contains only one simple hole (out of five different types) does not fold into a cube, (3) a complete characterisation when a rectangular polyomino with two or more unit square holes (but no other holes) can be folded into a cube, and (4) a sufficient condition when a simply-connected polyomino can be folded to a cube. These results answer several open problems of previous work and close the cases of tree-shaped polyominoes and rectangular polyominoes with just one simple hole.

翻译：哪些多联骨牌能够通过仅沿其下方正方形格子的边缘折痕，以±90°和±180°的折叠角度，并允许立方体面被多次覆盖，折叠成一个立方体？先前的研究针对树形多联骨牌和带孔多联骨牌进行了探讨，并给出了部分分类结果。我们证明存在一种算法可以判定给定多联骨牌能否折叠成立方体。该算法本质上需尝试所有可能的方式将多联骨牌的面映射至立方体的面，但（或许令人惊讶的是）验证此类映射是否对应有效折叠等价于拓扑学中的链环识别问题。我们还给出了关于可折叠或不可折叠成立方体的多联骨牌类别的进一步结果。我们的成果包括：（1）完整刻画所有能折叠成立方体的树形多联骨牌；（2）任何仅含一个简单孔洞（共五种类型）的矩形多联骨牌无法折叠成立方体；（3）对包含两个或多个单位正方形孔洞（无其他孔洞）的矩形多联骨牌能否折叠成立方体给出完整判定条件；（4）给出单连通多联骨牌可折叠成立方体的充分条件。这些结果回答了先前工作中的多个开放问题，并最终解决了树形多联骨牌及仅含一个简单孔洞的矩形多联骨牌的折叠判定问题。