Braid equivalences and the $L$--moves

TL;DR

This survey explores the use of $L$--moves to establish braid equivalences across various knot types and diagrammatic settings, providing a unified approach for geometric and algebraic proofs.

Contribution

It introduces and discusses the adaptation of $L$--moves as a universal tool for proving braid equivalences in diverse knot theories and diagrammatic contexts.

Findings

$L$--moves unify braid equivalence proofs across multiple knot types.

The approach applies to classical, virtual, welded, and singular knots.

Provides a geometric and algebraic framework for braid isotopy.

Abstract

In this survey paper we present the $L$--moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3--manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The $L$--moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements being first geometric and then algebraic.