proofs.

A [*chainmail graph*]{} is defined as a planar graph, possibly with loops but without parallel edges; the edges (including loops) are supplied with integer weights. We denote by $w_{ij} = w_{ji}$ the weight of the edge joining vertices $i$ and $j$; $w_{ii}$ is the weight of the loop attached to the vertex $i$. If the edge $[ij]$ is missing then $w_{ij} = 0$ by definition.

There is a way (see [@Toronto]) to define for every chainmail graph $G$ a closed oriented $3$-manifold $\Manif(G)$; any closed oriented $3$-manifold is $\Manif(G)$ for some $G$ (which is not unique). To the chainmail graph $G$ with $n$ vertices one associates two $n \times
n$-matrices: the adjacency matrix $W(G) = (w_{ij})$ and the Laplace (better to say, Schroedinger) matrix $L(G) = (l_{ij})$ where $l_{ij} {\stackrel{\mbox{\scriptsize def}}{=}}w_{ij}$ for $i \ne j$ and $l_{ii} {\stackrel{\mbox{\scriptsize def}}{=}}w_{ii} - \sum_{j \ne i} w_{ij}$. If all $w_{ii} = 0$ (such $G$ is called a balanced graph) then $L(G)$ is the usual (symmetric, degenerate) Laplace matrix $\widehat{W}$ from Section \[SSec:Main\].

1.  The rank of the homology group $H_1(\Manif(G),\Integer)$ is equal to $\dim \name{Ker} L(G)$.

2.  If $L(G)$ is nondegenerate (so that $\Manif(G)$ is a rational homology sphere and $H_1(\Manif(G),\Integer)$ is 