Describing realizable Gauss diagrams using the concepts of parity or bipartate graphs

TL;DR

This paper critiques recent parity-based criteria for realizable Gauss diagrams, providing counterexamples, and establishes that bipartite graphs offer a correct framework for their characterization.

Contribution

It corrects previous parity-based descriptions and proves that bipartite graphs accurately characterize realizable Gauss diagrams.

Findings

Parity-based conditions are insufficient for realizability.

Counterexamples disprove previous parity criteria.

Bipartite graphs correctly characterize realizable Gauss diagrams.

Abstract

Two recent publications describe realizable Gauss diagrams using conditions stating that the number of chords in certain sets of chords is even or odd. We demonstrate that these descriptions are incorrect by finding multiple counter-examples. However, the idea of having a parity-based description of realizable Gauss diagrams is attractive. We recall that realizability of Gauss diagrams as touch curves can be described via bipartite graphs. We show that realizable Gauss diagrams can be described via bipartite graphs.