We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number).

中文翻译：

---

不耐打结的硬度

我们证明，决定最多是否可以使用k个Reidemeister移动（其中k是输入的一部分）来解开小结的图是NP- hard。我们还证明了有关3球体中链接的几个自然问题都是NP难题，包括检测链接是否包含具有n个分量的琐碎子链接，计算链接的未链接数以及计算与四个相关的各种链接不变量维拓扑（例如4球欧拉特性，切片数和4维扣环数）。