On link diagrams that are minimal with respect to Reidemeister moves I and II

TL;DR

This paper proves the uniqueness of minimal link diagrams under Reidemeister moves I and II for links without trivial split components, and explores the implications for minimal diagrams and Reidemeister move III.

Contribution

It establishes the uniqueness of minimal diagrams under certain moves and characterizes the effects of Reidemeister move III on minimal diagrams.

Findings

Minimal diagrams are unique for links without trivial split components.

Every link has infinitely many minimal diagrams due to connected sums.

Reidemeister move III either preserves or simplifies to a special move regarding minimal diagrams.

Abstract

In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof also shows that the number of crossings of such a minimal diagram is unique for any diagram of any link. As the unknot admits infinitely many non-trivial minimal diagrams, we see that every link has infinitely many minimal diagrams, by considering the connected sums with such diagrams. We show that for a link without a trivial split component, an arbitrary Reidemeister move III either does not change the associated minimal diagram or can be reduced to a special type of a move up to Reidemeister moves I and II.