The unknotting numbers for plus-welded knotoids

TL;DR

This paper explores unknotting numbers for plus-welded knotoids, establishing transformation moves, extending warping degree, and providing upper bounds for unknotting numbers using crossing change and virtualization.

Contribution

It introduces new unknotting operations and bounds for plus-welded knotoids, extending knot invariants and transformation techniques to this generalized setting.

Findings

Descending diagrams can be simplified to trivial knotoids via specific moves.

Warping degree is extended and analyzed for plus-welded knotoids.

Upper bounds for unknotting numbers are established using crossing change and virtualization.

Abstract

Knotoid theory is a generalization of knot theory introduced by Turaev in 2012. In recent years, various invariants of knotoids have been studied. In this paper, we mainly discuss unknotting moves and unknotting numbers of plus-welded knotoids. Firstly, we prove that a descending diagram of a plus-welded knotoid can be transformed into a trivial one through a finite sequence of $\Omega_1$, $V\Omega_1 - V\Omega_4$, $\Omega_v$, $\Phi_{\text{over}}$, and $\Phi_+$-moves. Secondly, we extend the warping degree of knots to plus-welded knotoids and discuss its properties. Finally, by utilizing the descending diagram and the warping degree, we obtain two unknotting operations for plus-welded knotoids, referred as a crossing change and a crossing virtualization. For both operations, we find upper bounds for corresponding unknotting numbers of plus-welded knotoids.