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Complexity of unknotting of trivial 2-knots
Journal of Topology and Analysis ( IF 0.5 ) Pub Date : 2019-09-30 , DOI: 10.1142/s1793525320500272 Boris Lishak 1 , Alexander Nabutovsky 2
Journal of Topology and Analysis ( IF 0.5 ) Pub Date : 2019-09-30 , DOI: 10.1142/s1793525320500272 Boris Lishak 1 , Alexander Nabutovsky 2
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We construct a family of trivial 2 -knots k i in ℝ 4 such that the maximal complexity of 2 -knots in any isotopy connecting k i with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of k i .
Here, we can either construct k i as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots k i , 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 ℝ 3 , 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:
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ℝ 4 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ℝ 3 ,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。
更新日期:2019-09-30
中文翻译:
平凡的 2-knots 解开的复杂性
我们构建了一个平凡的家庭