Knots that are not slice either in positons or in negatons

TL;DR

This paper demonstrates the existence of infinitely many knots that cannot be unknotted through only positive or only negative crossing changes, by analyzing their bounding properties in certain 4-manifolds.

Contribution

It introduces the concept of knots that do not bound null-homologous disks in positons or negatons, revealing new obstructions in knot theory.

Findings

Existence of infinitely many such knots.

Knots that cannot be unknotted by only positive crossing changes.

Knots that cannot be unknotted by only negative crossing changes.

Abstract

An oriented compact 4-manifold $V$ with boundary $S^3$ is called a positon (resp. negaton) if its intersection form is positive definite (resp. negative definite) and it is simply connected. In this paper, we prove that there exist infinitely many knots which cannot bound null-homologous disks either in positons or in negatons. As a consequence, we find knots that cannot be unknotted either by only positive crossing changes or by only negative crossing changes.