Gauss diagrams as cubic graphs: The choice of the Hamiltonian cycle matters

TL;DR

This paper investigates how the choice of Hamiltonian cycle influences the properties of Gauss diagrams, demonstrating that certain cycle changes preserve realizability and illustrating the impact on diagram realizability and correspondence to curves.

Contribution

It provides examples showing the effect of Hamiltonian cycle choice on Gauss diagram properties and proves conditions under which realizability is preserved.

Findings

Changing the Hamiltonian cycle can alter Gauss diagram realizability.

Certain natural cycle changes preserve realizability.

Examples demonstrate differences in diagrams based on cycle choice.

Abstract

We explore to what extent the properties of a Gauss diagram are affected by the choice of its Hamiltonian cycle. We present an example of a realizable Gauss diagram and an unrealizable Gauss diagram that differ only by a choice of the Hamiltonian cycle. We present an example of two Gauss diagrams that correspond to different curves and differ only by a choice of the Hamiltonian cycle. We prove that a certain natural type of change of the Hamiltonian cycle preserves the realizability of the Gauss diagram.