  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.
