Curves on surfaces and moduli of associative algebras

Given an immersion of a circle in a punctured surface $\mathrm{\Sigma }$, we give an explicit (and finite) computation of the $A_{\mathrm{\infty }}$-algebra associated with this curve when viewed as an object in a (relative) Fukaya category of $\mathrm{\Sigma }$ in terms of the signed Gauss word recording the double points in a traversal of the curve and the visible polygons that it bounds in $\mathrm{\Sigma }$. We illustrate our computational technique by fully determining the $A_{\mathrm{\infty }}$-products for immersions with up to three self-intersections. In particular, it is proved that, over an algebraically closed field, all associative algebras of dimension $\le 4$, with one exception, can be realized as the (degree 0) endomorphism algebra of some Lagrangian immersion of a circle equipped with a bounding cochain computed in some relative Fukaya category $\mathcal{F}(\mathrm{\Sigma },D)$. We also note that any finite-dimensional algebra with radical square zero arises as the (degree 0) endomorphism algebra of an object in the Fukaya category $\mathcal{F}(\mathrm{\Sigma })$ of some punctured surface $\mathrm{\Sigma }$.