Minimal generating sets of rotational Reidemeister moves

TL;DR

This paper identifies minimal generating sets of rotational Reidemeister moves for tangle diagrams, establishing the exact number of moves needed for unframed and framed cases, which advances understanding of quantum invariants.

Contribution

It provides a detailed description and proof of the minimal number of rotational Reidemeister moves required to generate all such moves for unframed and framed tangles.

Findings

Minimal 8 moves generate all unframed rotational Reidemeister moves.

Minimal 9 moves generate all framed rotational Reidemeister moves.

A minimal generating set for framed links contains 5 moves.

Abstract

Rotational tangle diagrams have been proven to be extremely important in the study of quantum invariants, as they provide a natural passage between topology and quantum algebra. In this paper, we give a detailed description of several generating sets of rotational Reidemeister moves for rotational diagrams of both unframed and framed tangles. In particular, we prove that the minimal number of moves needed to generate all oriented unframed (resp. framed) rotational Reidemeister moves is 8 (resp. 9). The latter implies that a minimal generating set of Reidemeister moves for oriented, framed links contains 5 moves.