I am studying now the article of V. O. Manturov about free knots ("Free Knots and Parity", https://arxiv.org/pdf/0912.5348.pdf). I got stuck on the section "Parity as homology" (p.6), at the homology group $H_1(\Gamma,\mathbb{Z}_2)$. I don't know how, is it defined? Thought first that this is simply the homology of the graph understood as a 1-dimensional CW-complex. With this definition, I don't know, however, how to interpret this sentence:
"The homology group $H_1(\Gamma,\mathbb{Z}_2)$ is generated by "halves" corresponding to vertices: for every vertex $v$ we have two halves of the graph $\Gamma_{v,1}$ and $\Gamma_{v,2}$, obtained by smoothing at this vertex".