将多联骨牌折叠成立方体
Folding polyominoes into cubes
AI总结：
本文研究多联骨牌折叠成立方体的判定问题，提出一种基于面映射与解链识别的算法，并完整刻画了树形多联骨牌及若干带孔矩形多联骨牌的可折叠性。
摘要：
哪些多联骨牌可以折叠成一个立方体，其中只允许多联骨牌下方方格网的边作为折痕，折叠角度为 $\pm 90^\circ$ 和 $\pm 180^\circ$，并且允许立方体的面被覆盖多次？此前的结果研究了树形多联骨牌和带孔多联骨牌，并对这些情形给出了部分分类。我们证明存在一个算法，可以判定给定的多联骨牌能否折叠成立方体。该算法本质上相当于尝试将多联骨牌的面映射到立方体面的所有可能方式，但（或许令人惊讶的是）检查这种映射是否对应于一个有效折叠等价于拓扑学中的解链识别问题。我们还给出了关于哪些多联骨牌类可以或不可以折叠成立方体的进一步结果。我们的结果包括：（1）对所有可以折叠成立方体的树形多联骨牌的完整刻画；（2）任何只包含一个简单孔（五种不同类型之一）的矩形多联骨牌都不能折叠成立方体；（3）完整刻画了具有两个或更多单位方格孔（但没有其他孔）的矩形多联骨牌何时可以折叠成立方体；（4）给出了单连通多联骨牌可以折叠成立方体的一个充分条件。这些结果回答了先前工作中的若干开放问题，并解决了树形多联骨牌和仅有一个简单孔的矩形多联骨牌的情形。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
