Multiflypes of rectangular diagrams of links

TL;DR

This paper introduces a new large family of transformations for rectangular diagrams of links that preserve isotopy, demonstrating their ability to relate diagrams of equal complexity beyond simpler moves.

Contribution

The authors present a novel class of transformations for rectangular link diagrams that preserve isotopy and can relate diagrams not connected by simpler moves.

Findings

New transformations preserve link isotopy.

Existence of diagrams related by these transformations but not by simpler moves.

Transformations can relate diagrams of the same complexity.

Abstract

We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of `simpler' moves not increasing the complexity of the diagram along the way.