Title: $G 1$-smooth splines on quad meshes with 4-split macro-patch elements,
Abstract: We analyze the space of differentiable functions on a quad-mesh $\cM$, which
are composed of 4-split spline macro-patch elements on each quadrangular face.
We describe explicit transition maps across shared edges, that satisfy
conditions which ensure that the space of differentiable functions is ample on
a quad-mesh of arbitrary topology. These transition maps define a finite
dimensional vector space of $G^{1}$ spline functions of bi-degree $\le (k,k)$
on each quadrangular face of $\cM$. We determine the dimension of this space of
$G^{1}$ spline functions for $k$ big enough and provide explicit constructions
of basis functions attached respectively to vertices, edges and faces. This
construction requires the analysis of the module of syzygies of univariate
b-spline functions with b-spline function coefficients. New results on their
generators and dimensions are provided. Examples of bases of $G^{1}$ splines of
small degree for simple topological surfaces are detailed and illustrated by
parametric surface constructions.