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arXiv · 1510.02773

Complexity of Unknotting of Trivial 2-knots

Boris Lishak·Alexander Nabutovsky
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Abstract

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ 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 $\mathbb{R}^3$, where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.

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Boris Lishak, Alexander Nabutovsky. 2019-12-15. Complexity of Unknotting of Trivial 2-knots. https://doi.org/10.1142/s1793525320500272

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