Characterization of a non-trivial non-peripheral element of the free homotopy classes of a compact bordered surface Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of
curves in $\Sigma$. We say $x\in \widehat \pi(\Sigma)$ is peripheral, if there is representative $\alpha\colon \Bbb S^1\to \Sigma$ of $x$ with $\text{im}(\alpha)\subseteq \partial \Sigma$.

Question: Let $x\in\widehat \pi(\Sigma)$ be a non-trivial non-peripheral element. Does there exist a simple closed curve
$\beta\subseteq \Sigma$ such that the geometric intersection of
$\alpha$ and $\beta$ is non-zero, where $\alpha$ is a representative
of $x$.

 A: I don't think so. Let $\Sigma$ be the pair of pants, and $\alpha$ the curve, both pictured below. Then $x=[\alpha]$ is non-trivial since it has non-zero winding number with one of the holes. It is non-peripheral because it has non-zero winding number with two holes. Now, $x$ is trivial in the plane, so any loop $\beta$ in $\Sigma$, simple or not, wil have zero intersection number with any representative of $x$.

A: This is true for all (compact, connected, oriented) surfaces that admit essential non-peripheral simple closed curves.
The sphere, disk, annulus and pants do not admit essential non-peripheral curves.  The sphere and disk have trivial fundamental group.  So the answer is "yes" vacuously for the sphere and disk, and "no" for the annulus and pants (as suggested by Johannes’).
Suppose that $S$ is any other compact, connected, oriented surface.
Now $S$ admits filling laminations. Any sufficiently good (Hausdorff close) simple closed curve approximation to a filling lamination does what you want.  The existence of such laminations is a consequence of Thurston’s theory of pseudo-Anosov maps.
