## 11 Multivariate Splines and Cofactor Matroids
### Coning Splines: Abstract 4-Dimensional Rigidity Matroids and Multivariate Splines  
First we note that the same reduction technique used above will reduce the columns for the \(C_{3}^{2}\)-cofactor matrix to 4 columns per vertex, with an overall kernel of dimension 10. It is conjectured in [21] that the \(C_{3}^{2}\)-cofactor matroid is the maximal abstract 4-dimensional rigidity matroid and some evidence is offered for this conjecture. For generic configurations, there are graphs, such as \(K_{6,6}\), which are known to be independent in the \(C_{3}^{2}\)-cofactor matroid and known to be dependent in the 4-dimensional generic rigidity matroid [35]. The \(C_{3}^{2}\)-cofactor matroid uses cubes for the line coefficients and hence we evade the trap of dependence which is guaranteed in the 4-dimensional generic rigidity matroid for select bipartite bar-joint frameworks imposed by quadric surfaces for distance in \(\mathbb{P}^{4}\). Since we are exploring projective techniques in this paper, we note two relevant forms of coning for these spline matroids [21, 23, 24].  
1. In ([21] Theorem 5.3) it was verified that coning transfers maximal rank and independence from the \(C_{2}^{1}\)-cofactor matroid to the \(C_{3}^{2}\)-cofactor matroid on graphs at generic configurations in the plane. This is support for the conjecture mentioned above. Our expectation is that all graphs shown to be independent in \(\mathbb{P}^{d}\) will be independent in \(C_{d-1}^{d-2}\). We propose it is appropriate to extend any analysis of independent sets in \(\mathbb{P}^{d}\) to also explore the same graphs in the corresponding spline matroid [155].  
2. Multivariate splines also offer a different coning up a spatial dimension from bivariate \(C_{s+1}^{s}\)-splines to trivariate \(C_{s+1}^{s}\)-splines [23]. Note the indices are unchanged in this coning. This is a geometric theorem and follows the exact pattern described for body-hinge frameworks in Section 10.3. In particular, the space of trivariate splines around a vertex in a 3-dimensional tetrahedral decomposition of a ball are isomorphic to the space of bivariate splines on a generalised triangulation of a disc in the plane (a triangulation where triangles may overlap) [23].  
This coning up in spatial dimension and projecting down from the central vertex of a vertex figure are dual. In particular, coning on a plane triangulation produces a vertex figure for a tetrahedral decomposition and projecting down creates what is now called a generalized plane triangulation, since the projected triangles can now overlap [23]. These operations open up the significance of the projective invariance of multivariate splines as a tool within approximation theory. In particular, when we can preserve properties and spaces while coning up, we can move the higher dimensional cone around in the higher dimensional space, and re-project to create a projective image of the original spline realisation. This process also applies to general \(C_{d}^{1}\)-cofactors showing their projective invariance and should extend to \(C_{d}^{r}\)-cofactors in arbitrary dimensions.  
Although not usually presented this way, the coning reminds us that we can transfer the theory of \(C_{d}^{r}\)-cofactors to a cone and onto splines presented over a decomposition of the sphere. There is future work to confirm what appears to be a natural transfer.