  A cluster algebra is unistructural if the set of its cluster variables
determines its clusters and seeds. It is conjectured that all cluster algebras
are unistructural. In this paper, we show that any cluster algebra arising from
a triangulation of a marked surface without punctures is unistructural. Our
proof relies on the existence of a positive basis known as the bracelet basis,
and on the skein relations. We also prove that a cluster algebra defined from a
disjoint union of quivers is unistructural if and only if the cluster algebras
defined from the connected components of the quiver are unistructural.
