An Affine Linear Solution for the 2-Face Colorable Gauss Code Problem in the Klein Bottle and a Quadratic System for Arbitrary Closed Surfaces

Let $\bar{P}$ be a sequence of length $2n$ in which each element of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ occurs twice. Let $P^{\mathrm{\prime }}$ be a closed curve in a closed surface $S$ having $n$ points of simple auto-intersections, inducing a 4-regular graph embedded in $S$ which is 2-face colorable. If the sequence of auto-intersections along $P^{\mathrm{\prime }}$ is given by $\bar{P}$, we say that is a {\em $P^{\mathrm{\prime }}$ 2-face colorable solution for the Gauss Code $\bar{P}$ on surface $S$} or a {\em lacet for $\bar{P}$ on $S$}. In this paper we present a necessary and sufficient condition yielding these solutions when $S$ is Klein bottle. The condition take the form of a system of $m$ linear equations in $2n$ variables over ${\mathbb{Z}}_2$, where $m\le n(n-1)\mathrm{/}2$. Our solution generalize solutions for the projective plane and on the sphere. In a strong way, the Klein bottle is an extremal case admitting an affine linear solution: we show that the similar problem on the torus and on surfaces of higher connectivity are modelled by a quadratic system of equations.