On long knots in the full torus

TL;DR

This paper develops new invariants for long knots in the full torus using picture-valued and free group-valued techniques, enhancing understanding of their relation to various topological objects.

Contribution

It introduces novel, easily comparable invariants for long knots in the full torus, linking them to Legendrian knots, knotoids, and 3-manifolds.

Findings

Invariants are powerful and easy to compare.

The techniques connect long knots to other topological objects.

The work extends previous methods to new classes of knots.

Abstract

The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian knots, knotoids, 3-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [6]. Long knots naturally appear in the study of classical knots [1, 8].