An unknotting invariant for welded knots

TL;DR

This paper introduces the unknotting twist number, an invariant for welded knots, and explores its properties, relationships with other invariants, and applications to Gordian distance and complexes.

Contribution

It defines the unknotting twist number for welded knots and relates it to existing invariants, providing new tools for studying welded knot complexity.

Findings

Unknotting twist number is an effective invariant for welded knots.

Lower bounds on twist number are established via Alexander quandle coloring.

Gordian distance and complex for welded knots are analyzed using twist moves.

Abstract

We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex.