Complexity of untying $2$-knots

Boris Lishak (Sydney)

Abstract

Embeddings of $S^2$ into $S^4$ can be topologically nontrivial. But even if the embedding is trivial, geometrically it can "look" knotted. We will give examples of topologically trivial $2$-knots such that their complexity will grow super-exponentially. Complexity will be either defined in terms of Riemannian geometry in the smooth case or combinatorially if the embedding is piece-wise linear. Proofs will use some group theory.