Gauss diagrams of real and virtual knots in the solid torus

TL;DR

This paper introduces a new type of Gauss diagrams for knots in the solid torus, demonstrating their effectiveness in reconstructing knot diagrams and characterizing decorated diagrams of closed braids.

Contribution

It defines a novel Gauss diagram framework for knots in the solid torus and applies it to recover diagrams and characterize closed braids.

Findings

Gauss diagrams can fully recover knot diagrams in the solid torus.

The new diagrams characterize decorated diagrams of closed braids.

Efficient tool for studying knots in the solid torus.

Abstract

We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of closed braids.