Complexity of Unknotting of Trivial 2-knots

TL;DR

This paper demonstrates that trivial 2-knots in four-dimensional space can have isotopies to the standard unknot with complexity growing faster than any fixed-height exponential tower, contrasting with classical knots in three dimensions.

Contribution

It constructs families of trivial 2-knots with complexity growth surpassing exponential towers during isotopies, highlighting a stark difference from classical knot unknotting.

Findings

Complexity of 2-knots can grow faster than any fixed-height exponential tower during isotopy.

Constructs smooth and PL-knots with high complexity growth.

Contrasts with classical knots where complexity growth is polynomial.

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.