On a characterization of Gauss codes (Q1302047)

The traversal of a self-crossing closed plane curve, with points of multiplicity at most two and with only a finite number of multiple points, defines a double occurrence sequence, the Gauss code of the curve. Given a sequence \(S\), two symbols \(u,v\) are said to be interlaced in \(S\) if exactly one occurrence of \(v\) appears in \(S\) between the two occurences of \(u\). This relation gives rise to the interlacement graph of \(S\), \(\Lambda (S)\). \textit{P. Rosenstiehl} [C. R. Acad. Sci., Paris, Sér. A 283, 551-553 (1976; Zbl 0345.05130)] characterized Gauss codes by \(\Lambda (S)\). Introducing a switch operation, the authors give a new characterization of these sequences by \(\Lambda (S)\) and then deduce a simple self-contained proof of Rosenstiehl's characterization.