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In this paper we study the dimension of bivariate polynomial splines of mixedsmoothness on polygonal meshes. Here, "mixed smoothness" refers to the choiceof different orders of smoothness across different edges of the mesh. To studythe dimension of spaces of such splines, we use tools from Homological Algebra.These tools were first applied to the study of splines by Billera (1988). Usingthem, estimation of the spline space dimension amounts to the study of thegeneralized Billera-Schenck-Stillman complex for the spline space. Inparticular, when the homology in positions one and zero of this complex aretrivial, the dimension of the spline space can be computed combinatorially. Wecall such spline spaces "lower-acyclic." In this paper, starting from a splinespace which is lower-acyclic, we present sufficient conditions that ensure thatthe same will be true for the spline space obtained after relaxing thesmoothness requirements across a subset of the mesh edges. This general recipeis applied in a specific setting: meshes of arbitrary topologies. We show howour results can be used to compute the dimensions of spline spaces ontriangulations, polygonal meshes, and T-meshes with holes.

Counting the dimension of splines of mixed smoothness: A general recipe, and its application to meshes of arbitrary topologies