A new unknotting operation for classical and welded links

TL;DR

This paper introduces the diagonal move, a new unknotting operation that efficiently simplifies classical and welded knots, unifying existing methods and reducing the number of steps needed for knot unknotting.

Contribution

The paper presents the diagonal move as a novel, unifying unknotting operation applicable to both classical and welded knots, improving efficiency in knot simplification.

Findings

Diagonal move reduces unknotting steps compared to crossing changes.

Applicable to both classical and welded knots.

Provides a new tool for knot equivalence and simplification.

Abstract

Any knot diagram can be transformed into the unknot by a series of unknotting operations. This paper introduces the diagonal move, a novel unknotting operation that generalizes and unifies several existing moves. We prove that the diagonal move is an efficient unknotting operation for both classical and welded knots, demonstrating that any knot or link can be reduced to the unknot or unlink via a finite sequence of diagonal moves and Reidemeister moves. Additionally, we analyze the distance between knots under diagonal moves, showing that it often requires fewer operations than traditional crossing changes, and extend our results to welded knots, confirming the diagonal move's applicability in this broader setting. Our findings provide a powerful new tool for knot simplification and equivalence, advancing topological and combinatorial knot theory.