On Usual, Virtual and Welded knotted objects up to homotopy

TL;DR

This paper explores different classes of knotted objects and their equivalence relations, highlighting differences through Gauss diagram techniques, advancing understanding of virtual and welded knot theories.

Contribution

It introduces new distinctions among usual, virtual, and welded knotted objects using Gauss diagram formulas and analyzes their equivalence relations.

Findings

Differences between various knotted object classes are clarified.

Gauss diagram techniques are effective for analyzing knot equivalences.

Results highlight the complexity of virtual and welded knot theories.

Abstract

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.