Trace Embeddings from Zero Surgery Homeomorphisms

TL;DR

This paper investigates the slice properties of knots related to zero surgery homeomorphisms, ruling out certain knots as slice and analyzing the potential for constructing exotic 4-manifolds, while extending some techniques and confirming standardness of specific homotopy spheres.

Contribution

It demonstrates that specific knots are not slice using zero surgery techniques and extends these methods to an infinite family, clarifying limitations in constructing exotic 4-manifolds.

Findings

Certain knots are proven not to be slice.

Techniques extend to an infinite family of zero surgery homeomorphisms.

A family of homotopy spheres are shown to be standard.

Abstract

Manolescu and Piccirillo recently initiated a program to construct an exotic $S^4$ or $\# n \mathbb{CP}^2$ by using zero surgery homeomorphisms and Rasmussen's $s$-invariant. They find five knots that if any were slice, one could construct an exotic $S^4$ and disprove the Smooth $4$-dimensional Poincar\'e conjecture. We rule out this exciting possibility and show that these knots are not slice. To do this, we use a zero surgery homeomorphism to relate slice properties of two knots \textit{stably} after a connected sum with some $4$-manifold. Furthermore, we show that our techniques will extend to the entire infinite family of zero surgery homeomorphisms constructed by Manolescu and Piccirillo. However, our methods do not completely rule out the possibility of constructing an exotic $S^4$ or $\# n \mathbb{CP}^2$ as Manolescu and Piccirillo proposed. We explain the limits of these methods hoping this will inform and invite new attempts to construct an exotic $S^4$ or $\# n \mathbb{CP}^2$. We also show a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.