On ordered sequences for link diagrams with respect to Reidemeister moves I and III

TL;DR

The paper investigates the transformation of link diagrams via Reidemeister moves I and III, proving limitations of certain sequences and introducing the concept of I-generalized ordered sequences for simpler transformations.

Contribution

It introduces the concept of I-generalized ordered sequences and proves their sufficiency for transforming link diagrams under Reidemeister moves I and III.

Findings

Certain pairs of trivial knot diagrams cannot be transformed using specific Reidemeister move sequences.

Infinitely many pairs of diagrams require generalized sequences for transformation.

I-generalized ordered sequences always suffice for transformations involving moves I and III.

Abstract

We first prove that, infinitely many pairs of trivial knot diagrams that are transformed into each other by applying Reidemeister moves I and III are NOT transformed into each other by a sequence of the Reidemeister moves I that increase the number of crossings, followed by a sequence of Reidemeister moves III, followed by a sequence of the Reidemeister moves I that decrease the number of crossings. To create a simple sequence between link diagrams that are transformed into each other by applying finitely many Reidemeister moves I and III, we prove that the link diagrams are always transformed into each other by applying an I-generalized ordered sequence.