Unknotted gropes, Whitney towers, and doubly slicing knots

TL;DR

This paper investigates the structure and unknottedness of gropes and Whitney towers in four-dimensional space, introducing new concepts and constructions that relate to knot sliceness and fundamental groups.

Contribution

It introduces a notion of unknottedness for gropes and Whitney towers, proves stability under modifications, and constructs geometric bi-filtrations of knots related to double sliceness.

Findings

Unknottedness is preserved under certain modifications.

Constructed handlebody structures for exteriors of gropes and Whitney towers.

Bi-filtrations of knots do not stabilize at any stage.

Abstract

We study the structure of the exteriors of gropes and Whitney towers in dimension 4, focusing on their fundamental groups. In particular we introduce a notion of unknottedness of gropes and Whitney towers in the 4-sphere. We prove that various modifications of gropes and Whitney towers preserve the unknottedness and do not enlarge the fundamental group. We exhibit handlebody structures of the exteriors of gropes and Whitney towers constructed by earlier methods of Cochran, Teichner, Horn, and the first author, and use them to construct examples of unknotted gropes and Whitney towers. As an application, we introduce geometric bi-filtrations of knots which approximate the double sliceness in terms of unknotted gropes and Whitney towers. We prove that the bi-filtrations do not stabilize at any stage.