Abstract

We present three “hard” diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in ${\mathbb{S}}^2$. Two of them are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small “standard” example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in ${\mathbb{S}}^2$.

中文翻译：

---

解结的硬图

摘要

我们展示了未知结的三个“硬”图。他们需要（至少）三个额外的交叉点，然后才能通过 Reidemeister 移动简化为琐碎的结图${\mathbb{小号}}^{\mathrm{2个}}$. 其中两个是通过应用先前提出的方法构建的。它们的硬度证明使用了大量的计算资源。我们还确定，对于 Reidemeister 的移动，硬结图的任何小“标准”示例都需要一个以上的额外交叉点${\mathbb{小号}}^{\mathrm{2个}}$.