Arc shift move and region arc shift move for twisted knots

TL;DR

This work advances the study of unknotting operations for twisted knots by introducing the region arc shift move and establishing that it is an unknotting operation. It proves that for any $n\in\mathbb{N}$ there are twisted knots with arc shift number $A(K)=n$, and constructs two distinct unknotting sequences revealing sharp invariants. The paper defines the region arc shift number $RA(K)$ and the forbidden number $\tilde{F}(K)$, showing $RA(K)\le \tilde{F}(K)$ and deriving lower bounds on $\tilde{F}(K)$ through the polynomial invariant $Q(s,t)$ and the odd writhe $J(K)$, with parity-dependent refinements. Using the affine-index based polynomial $Q(s,t)$, the authors distinguish knot families and provide explicit closed forms for $Q$ in key constructions, highlighting how these invariants govern unknotting complexity in twisted knots.

Abstract

In this paper, we study the unknotting operation for twisted knots, called arc shift move. First, we find a family of twisted knots with arc shift number $n$ for any given $n \in \mathbb{N}$. Then we define a new unknotting operation, called the region arc shift move for twisted knots and find family of twisted knots whose region arc shift number is less than or equal to $n$ for any given $n \in \mathbb{N}$. Later, we explore bounds for region arc shift number and forbidden number.

Figures (30)

• Figure 1: Extended Reidemeister moves.
• Figure 2: Gauss code and Gauss diagram of twisted knots.
• Figure 3: Forbidden Moves of twisted knots and the corresponding Gauss diagrams.
• Figure 4: Forbidden move $T_4$.
• Figure 5: Arc $(a,b)$ and arc $(c,d)$.

Theorems & Definitions (45)

• Remark 2.1
• Theorem 2.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Definition 2.3
• Theorem 2.5
• Definition 2.4
• Theorem 2.6