Virtual Knotoids in Thickened Surfaces

TL;DR

The paper develops a geometric framework for virtual knotoids by interpreting them as virtual arcs in thickened surfaces, showing that virtual knotoid theory encompasses classical knotoids and extends to higher-genus surfaces via stable equivalence. It proves that every virtual knotoid has a unique irreducible representation on its minimal genus surface, generalizing Kuperberg’s irreducibility result for virtual knots. By establishing a precise correspondence between virtual knotoids in $S^2$ and stable knotoids in surfaces, and unifying the arc-based with the surface-based perspectives, the authors demonstrate that classical knotoid theory embeds properly into virtual knotoid theory. The results bridge diagrammatic and three-dimensional viewpoints, with potential implications for applications that utilize knotoids and their virtualizations in thickened surfaces. All key notions are formalized through virtual arcs, stabilization/destabilization, and abstract knotoid correspondences, yielding a robust, coordinate-free interpretation of virtual knotoids.

Abstract

In this paper, we give a geometric interpretation of virtual knotoids as arcs in thickened surfaces. Then we show that virtual knotoid theory is a generalization of classical knotoid theory. This gives a proof of a conjecture of Kauffman and the first author.

Figures (21)

• Figure 1: Examples of knotoid diagrams.
• Figure 2: Moves for knotoids.
• Figure 3: Product of a knotoid diagram in sphere and a knotoid diagram in torus.
• Figure 4: The underpass (on the left) and overpass (on the right) closures of a knotoid diagram.
• Figure 5: The vertex multiplication of theta-curves.

Theorems & Definitions (46)

• Theorem
• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Remark 2.1.1
• Theorem 2.2.1
• Definition 2.3.1
• Definition 2.3.2
• Theorem 2.3.1
• Definition 2.3.3