The unknotting numbers for plus-welded knotoids

TL;DR

Addresses the unknotting problem for plus-welded knotoids by extending warping degree, analyzing descending diagrams, and exploiting virtual closure to welded knots. It proves that a descending diagram can be simplified to trivial via a finite sequence of moves $Ω_1$, $VΩ_1-VΩ_4$, $Ω_v$, $Φ_{ ext{over}}$, and $Φ_+$, and defines $d(K)$ to bound unknotting numbers. It derives two unknotting operations—crossing changes and crossing virtualizations—with upper bounds $u(K)\le d(K)$ and $u_v(K)\le d(K)$, plus the bound $u(K)\le (\operatorname{cr}(K)-1)/2$ for nontrivial cases, and discusses implications for Gordian-type structures via virtual closure to welded knots.

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 $Ω_1$, $VΩ_1 - VΩ_4$, $Ω_v$, $Φ_{\text{over}}$, and $Φ_+$-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.

Figures (25)

• Figure 1: Examples of knotoid diagrams.
• Figure 2: Classical Reidemeister moves.
• Figure 3: Forbidden knotoid moves.
• Figure 4: Classical crossing and virtual crossing.
• Figure 5: Virtual Reidemeister moves.

Theorems & Definitions (53)

• Definition 2.1
• Remark 2.1
• Definition 3.1
• Theorem 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Theorem 3.1
• Example 3.1