Unknotting rectangular diagrams of the trivial knot by exchanging moves

TL;DR

This paper demonstrates that rectangular diagrams of the trivial knot can be simplified to a crossing-free form solely through exchange moves, eliminating the need for merge operations.

Contribution

It proves that merge operations are unnecessary for deforming trivial knot diagrams into crossing-free forms, simplifying previous methods.

Findings

Trivial knot diagrams can be simplified using only exchange moves.

Merge operations are not required for trivial knot simplification.

Simplification process preserves the number of vertical edges.

Abstract

If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need no merge operations to deform a rectangular diagram of the trivial knot to one with no crossings.