[Math] Definition of the algebraic intersection number of oriented closed curves.

Question

Can anyone point me to a reference (book/paper) where I can read up on the
the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is
used to prove Proposition 2.7. The only resource where I could find something is Farb and Margalit: A primer on MCGs. It says there on page 28:

Let $\alpha,\beta$ be two oriented, transverse, simple closed curves on an oriented surface $S$. The algebraic intersection number $\hat{i}(\alpha,\beta)$ is defined as the sum of the indeces of the intersection points of $\alpha$ and $\beta$, where an index of an intersection point is $+1$ if the orientation of the intersection agrees with the orientation of $S$ and $-1$ otherwise.

1. First of all, I can't really make sense of the notion of agreement of the orientation of an intersection with the orientation of the surface. What I could imagine is choosing one of the curves ($\alpha$) as a reference and counting an intersetion with $+1$ if the other curve ($\beta$) crosses $\alpha$ from left to right and with $-1$ if $\beta$ crosses $\alpha$ from right to left.

2. In Farb and Margalit's book it is mentioned that this intersection number is invariant for homologically equivalent loops. Does anyone know where I can find a proof?

Answer 1

Perhaps a clearer way of saying it is that if two oriented submanifolds $N_1, N_2$ intersect transversaly at a point $x$ of an oriented manifold $M$, then their intersection number is $+1$ if first a direct basis of $T_x N_1$, then a direct basis of $T_xN_2$ forms a direct basis of $T_xM$.