From planar to annular to toroidal bracket polynomials for pseudo knots and links

TL;DR

This paper extends classical polynomial invariants from planar pseudo links to annular and toroidal pseudo links, providing new tools for their study in different topological settings.

Contribution

It introduces generalized Kauffman bracket and Jones-type polynomials for annular and toroidal pseudo links and their lifts, expanding the scope of link invariants.

Findings

Defined new polynomial invariants for annular and toroidal pseudo links.

Extended invariants to mixed links in the three-sphere.

Provided a framework for analyzing pseudo links in various topological contexts.

Abstract

Pseudo links are equivalence classes under Reidemeister-type moves of link diagrams containing crossings with undefined over and under information. In this paper, we extend the Kauffman bracket and Jones-type polynomials from planar pseudo links to annular and toroidal pseudo links and their respective lifts from the three-space to the solid torus and the thickened torus. Moreover, since annular and toroidal pseudo links can be represented as mixed links in the three-sphere, we also introduce the respective Kauffman bracket and Jones-type polynomials for their planar mixed link diagrams. Our work provides new tools for the study of annular and toroidal pseudo links.