Smoothly knotted surfaces that remain distinct after many internal stabilizations

TL;DR

This paper demonstrates that in 4-manifolds, pairs of smoothly knotted surfaces can require arbitrarily many internal stabilizations to become isotopic, revealing complex behaviors in surface knotting and stabilization.

Contribution

It introduces the concept of subtly smoothly knotted surfaces and shows that many such surfaces have unbounded stabilization distances, expanding understanding of surface isotopy classes.

Findings

No upper bound on internal stabilization steps needed for isotopy.

Many surfaces can be modified to have large stabilization distance.

Stabilizing 4-manifolds can produce non-isotopic, smoothly related copies of 3-manifolds with positive first Betti number.

Abstract

Internal stabilization adds a trivial handle to an embedded surface in a coordinate chart. It is known that any pair of smoothly knotted surfaces in a simply-connected $4$-manifold become smoothly isotopic after sufficiently many internal stabilizations. In this paper, we show that there is no upper bound on the number of internal stabilizations required. In fact, this behavior is fairly generic. The definition of a subtly smoothly knotted pair of surfaces is given and it is shown that many surfaces may be modified to obtain subtly knotted surfaces with large internal stabilization distance. Furthermore, it is shown that after stabilizing any $4$-manifold with contain topologically isotopic, smoothly related, non-isotopic copies of any $3$-manifold having a positive first betti number.