On Realizability of Gauss Diagrams

TL;DR

This paper investigates the realizability of Gauss diagrams by knots, providing a direct approach and conditions based on entrance-exit counts and the Jordan curve theorem.

Contribution

It introduces a new direct method for determining Gauss diagram realizability using entrance-exit counts and topological principles.

Findings

Realizability conditions relate to equal numbers of entrances and exits.

Sufficient conditions are derived from the Jordan curve theorem.

The approach simplifies previous methods for Gauss diagram realization.

Abstract

The problem of which Gauss diagram can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entrances" and the sufficient condition is based on Jordan curve Theorem.