The Wikipedia article on the FKT algorithm says that one finds the number of perfect matchings in an undirected planar graph $G$ as follows.
Find a graph $H$ that is a directed version of $G$, such that an odd number of edges are oriented clockwise for every face in $G$.
Calculate $B$, the $(1, −1, 0)$-adjacency matrix of $H$.
The perfect number of matchings for $G$ is $\rm{pf}(B) = \sqrt{\det(B)}$.
The hard part is step 1, which is what FKT deals with.
I have two questions:
Is my above summary correct?
What does it mean by the $(1, −1, 0)$-adjacency matrix? From the article it links to, I would guess that you put $1$ for each edge, $-1$ for the lack of an edge and $0$ on the diagonal. But this would not yield a skew symmetric $B$. Is it instead $1$ along the direction of an edge, $-1$ against, and $0$ for none?
Assume that the vertex set of $G$ is $\{1,\ldots,n\}$. If $i$ and $j$ are not adjacent, set $B_{i,j}=0$. If $i$ and $j$ are adjacent and $i<j$, set $B_{i,j}=+1$ and $B_{j,i}=-1$; if $i$ and $j$ are adjacent and $i>j$ set $B_{i,j}=-1$ and $B_{j,i}=+1$. Now $B$ is skew symmetric and $\det(B)$ is the square of the number of perfect matchings of $G$.