We construct a family of trivial 2-knots ki in ℝ4 such that the maximal complexity of 2-knots in any isotopy connecting ki with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of ki. Here, we can either construct ki as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots ki, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat 2-simplices in its triangulation. These results contrast with the situation of classical knots in ℝ3, where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.

中文翻译：

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平凡的 2-knots 解开的复杂性

我们构建了一个平凡的家庭2-结ķ一世在ℝ4使得最大复杂度2-任何同位素连接中的结ķ一世与标准的 unknot 相比，复杂度的任何固定高度的指数塔增长得更快ķ一世. 在这里，我们可以构造ķ一世作为平滑嵌入并测量它们的复杂性作为ropelength（又名皱缩度）或构建PL-knotsķ一世，考虑通过 PL 结的同位素，并将 PL 结的复杂性测量为平面的最小数量2- 简化了它的三角剖分。这些结果与经典结的情况相反ℝ3，其中每个未结都可以通过复杂度仅比初始结的复杂度高多项式的结来解开。