Complexity of unknotting of trivial 2-knots

Abstract

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