Satellite knots and trivializing bands

TL;DR

This paper constructs an infinite family of satellite knots that can be unknotted with a single band move, but no such move can be disjoint from the satellite torus, revealing new insights into knot unknotting processes.

Contribution

It introduces an infinite family of satellite knots with a unique unknotting property related to band moves and the satellite torus.

Findings

Existence of satellite knots unknottable by a single band move

No band move disjoint from the satellite torus can unknot these knots

Provides new understanding of the relationship between satellite structure and unknotting operations

Abstract

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.