From annular to toroidal pseudo knots

TL;DR

This paper extends pseudo knot theory from planar cases to annular and toroidal cases, introducing new invariants and exploring their relationships within these more complex topological settings.

Contribution

It introduces the theories of annular and toroidal pseudo knots, their representations, and new invariants for classifying pseudo knots in these contexts.

Findings

Extended pseudo knot theory to annular and toroidal cases.

Represented these knots as planar mixed pseudo links.

Defined new invariants for classifying pseudo knots in solid and thickened tori.

Abstract

In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar ${\rm O}$-mixed and ${\rm H}$-mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of ${\rm O}$-mixed and ${\rm H}$-mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus.