A new unknotting operation for classical and welded links
Danish Ali, Zhiqing Yang, Mohd Ibrahim Sheikh, Sidra Batool
TL;DR
This work introduces the $D$-move, a local diagram transformation that generalizes and unifies several classic unknotting moves. It proves that any knot or link diagram can be reduced to the trivial diagram using finite sequences of $D$-moves together with Reidemeister moves, and defines the diagonal unknotting number $u_D(K)$. The paper shows that numerous known moves, including the $\Delta$-move, $\sharp$-move, $\Gamma$-move, $n$-gon move, and pass move, are realizable by $D$-moves, yielding distance relations $d_{D}(K,K') \le d_{X}(K,K') \le 2 d_{D}(K,K')$ and positioning $D$-move as a powerful generalization. It then extends these results to welded knots, proving that the $D$-move remains an unknotting operation and that all classical moves are expressible via $D$-moves, establishing $D$-equivalence for welded knots. Collectively, these results provide a unified, efficient framework for knot simplification and equivalence with potential broad implications in topological and combinatorial knot theory.
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.