2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229738By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.15 pages, 4 figures. Second version: added reference, corrected some commentsGeometric Topology57M99 (Primary); 68R15 (Secondary)text