  We study the dimension and construct a basis for $C^1$-smooth isogeometric
function spaces over two-patch domains. In this context, an isogeometric
function is a function defined on a B-spline domain, whose graph surface also
has a B-spline representation. We consider constructions along one interface
between two patches. We restrict ourselves to a special case of planar B-spline
patches of bidegree (p,p) with $p \geq 3$, so-called analysis-suitable $G^1$
geometries, which are derived from a specific geometric continuity condition.
This class of two-patch geometries is exactly the one which allows, under
certain additional assumptions, $C^1$ isogeometric spaces with optimal
approximation properties (Collin et al., 2016).
  Such spaces are of interest when solving numerically fourth-order PDE
problems, such as the biharmonic equation, using the isogeometric method. In
particular, we analyze the dimension of the $C^1$-smooth isogeometric space and
present an explicit representation for a basis of this space. Both the
dimension of the space and the basis functions along the common interface
depend on the considered two-patch parameterization. Such an explicit, geometry
dependent basis construction is important for an efficient implementation of
the isogeometric method. The stability of the constructed basis is numerically
confirmed for an example configuration.
