Deformations of Gentle $A_{\mathrm{\infty }}$-Algebras

In this paper we calculate the Hochschild cohomology of gentle $A_{\mathrm{\infty }}$-algebras of arc collections on marked surfaces without boundary components. When the underlying arc collection has no loops or two-cycles, we show that the dgla structure of the Hochschild complex is formal and give an explicit realization of all deformations up to gauge equivalence.