Classification of virtual links by arc shift move

Abstract

In this paper, we establish that the arc shift operation on a $n$-component virtual link diagram acts as an unknotting operation when the virtual link is $n$-homogeneous proper, aiding in the classification of \( n \)-component virtual links up to arc shift equivalence. We explore the connection between the arc shift number and the odd writhe of virtual links which are homogeneous proper. Additionally, we identify sequences of virtual link diagrams \( L_n \) for which the upper bound of the arc shift number is exactly \( n \).

Figures (23)

• Figure 1: Generalized Reidemeister moves for virtual link diagrams.
• Figure 2: Detour move.
• Figure 3: Signs of crossing points: $sgn(c)=+1$ and $sgn(c')=-1$.
• Figure 4: Gauss diagram under generalized Reidemeister moves.
• Figure 5: Gauss diagram for virtual link.

Theorems & Definitions (60)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Theorem 2.6
• Theorem 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10