Essential tori in link complements: impossibility of detecting the satellite structure by monotonic simplification

TL;DR

This paper investigates whether satellite structures in link complements can always be detected through monotonic simplification of rectangular diagrams, and demonstrates that this is not always possible.

Contribution

It proves that monotonic simplification cannot always reveal satellite structures in link diagrams, highlighting limitations of this method.

Findings

Monotonic simplification fails to detect satellite structures in some cases.

The paper provides a counterexample to the conjecture that all satellite links can be simplified to reveal their structure.

It advances understanding of the limitations of diagram simplification techniques in knot theory.

Abstract

In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question.