Complexity of Unknotting of Trivial 2-knots

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.

References in corpus (2)

• Doubly slice knots with low crossing number
• Balanced presentations of the trivial group and four-dimensional geometry

Cited by in corpus (1)

• Linear embeddings of contractible and collapsible complexes