Suppose that we have two smooth, simple closed curves $\alpha, \beta : S^{1} \rightarrow \Sigma$ in a closed, connected, oriented surface $\Sigma$, and suppose that $\alpha \cap \beta$ consists of finitely many points, all of which are transverse intersections. Then we can define the oriented intersection number $i(\alpha,\beta)$ of $\alpha$ and $\beta$ as the sum of oriented intersection numbers $i_{p}(\alpha,\beta)$ at the intersection points $p$, where $i_{p}(\alpha,\beta)$ is $\pm 1$ depending on whether the crossing $\alpha,\beta$ is positive or negative with respect to the orientation on $\Sigma$.
The oriented intersection number $i(\alpha,\beta)$ can be computed algebraically in terms of Poincar'e duality. Namely, let $[\alpha],[\beta] \in H_{1}(\Sigma)$ be the homology classes, $[\alpha]^{\vee},[\beta]^{\vee} \in H^{1}(\Sigma)$ the Poincar'e dual cohomology classes (duality being with respect to the fixed orientation). Then $$i(\alpha,\beta)=[\alpha]^{\vee} \cup [\beta]^{\vee} \in H^{2}(\Sigma) \simeq H_{0}(\Sigma) \simeq Z.$$ (Here we have used duality again.)
I'd like to generalize this kind of calculation to intersections of smooth but not necessarily closed curves in a connected, oriented surface $(\Sigma,\partial \Sigma)$ with boundary, where the end points of non-closed curves are required to lie in the boundary $\partial \Sigma$ and the curves don't intersect on the boundary. The sum $i(\alpha,\beta)$ over signed intersection numbers still makes sense, but it's unclear to me what invariance properties it has, and whether it can be computed algebraically in terms of Leftschetz duality.
Questions:
In the situation above with boundary, is it true that $i(\alpha,\beta)$ is invariant under homotopy/isotopy relative to the boundary (that is, fixing the boundary pointwise)?
Can $i(\alpha,\beta)$ be computed "algebraically", maybe using Lefschetz duality?
If the above are true, are there textbook or research paper references?
Concerning 2), what I tried and what didn't work is the following:
Suppose $\alpha, \beta$ are non-closed curves with end points in the boundary $\partial \Sigma$. Then they define relative homology classes in $H_{1}(\Sigma,\partial \Sigma) \simeq H^{1}(\Sigma)$, where the last isomorphism is Lefschetz duality. But now if you try to cup together the dual classes, you get a class in $H^{2}(\Sigma)$, which is zero because $\Sigma$ is not closed. So this is useless.