A homological solution for the Gauss code problem in arbitrary surfaces (Q2483476)

A Gauss code \(P\) is a cyclic sequence in the set \(E=\{1,\dots,n\}\) in which each element in \(E\) occurs twice. Let \(P'\) be a closed curve in a closed surface \(S\) having \(n\) points of simple self-intersection. This induces a 4-regular graph embedded into \(S\) such that the cyclic sequence of self-intersections reproduces \(P\). It is said that \(P'\) is a lacet of \(P\) on \(S\) if the embedding of \(P'\) produces a 2-colorable \(t\)-map. In this paper, the authors give characterizations for the realizations of lacets in the torus and in the Klein bottle in terms of the existence of a pair of \(0-1\) vectors with certain properties. Also an algorithmic solution for the Klein bottle is obtained.