A construction of slice knots via annulus modifications

TL;DR

This paper introduces n-twist annulus modifications as a new operation on homology 4-balls, providing novel constructions of slice and exotically slice knots, and establishing connections with existing techniques like annulus twists.

Contribution

It defines n-twist annulus modifications and demonstrates their use in constructing and relating slice knots, including exotically slice knots, with new examples and broader applicability.

Findings

Constructed new smoothly slice knots with non-slice derivatives.

Connected n-twist annulus modifications to existing annulus twist techniques.

Showed any exotically slice knot can be obtained via annulus modifications from the unknot.

Abstract

We define an operation on homology ${B}^4$ which we call an $n$-twist annulus modification. We give a new construction of smoothly slice knots and exotically slice knots via $n$-twist annulus modifications. As an application, we present a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. Also, we relate $n$-twist annulus modifications to $n$-fold annulus twists which was first introduced by Osoinach, then has been used by Abe and Tange to construct smoothly slice knots. Furthermore we consider $n$-twist annulus modifications in more general setting to show that any exotically slice knot can be obtained by the image of the unknot in the boundary of a smooth $4$-manifold homeomorphic to ${B}^4$ after an annulus modification.