A short proof of the equivalence of any Reidemeister oriented move 3

TL;DR

This paper provides a concise proof that a minimal set of Reidemeister moves, specifically type 2 moves plus one type 3 move, can generate all other oriented Reidemeister moves, simplifying the understanding of knot diagram transformations.

Contribution

It offers a short, elegant proof demonstrating the sufficiency of a small move set to derive all Reidemeister moves, streamlining knot theory techniques.

Findings

Type 2 moves plus one type 3 move imply all other type 3 moves.

Simplifies the set of moves needed for knot diagram transformations.

Provides a more efficient framework for studying knot invariants.

Abstract

In this note we present a short proof that the 4 oriented Reidemeister moves of type 2 together with any one of the 8 oriented Reidemeister moves of type 3 are sufficient to imply the other 7.