Extensions of some classical local moves on knot diagrams

TL;DR

This paper explores various local moves on knot diagrams, providing algebraic classifications, relationships, and welded extensions, demonstrating their role as unknotting operations for welded knots.

Contribution

It offers a comprehensive algebraic classification of classical and welded local moves and investigates their extensions and topological interpretations.

Findings

All considered moves are unknotting operations for welded long knots.

Classical quotient embeds into welded quotient for these moves.

Provides algebraic and topological insights into local moves on knots.

Abstract

In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a by-product, we obtain that all of the above local moves are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.