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A conjecture of Graver from 1991 states that the generic $3$-dimensionalrigidity matroid is the unique maximal abstract $3$-rigidity matroid withrespect to the weak order on matroids. Based on a close similarity between thegeneric $d$-dimensional rigidity matroid and the generic$C_{d-2}^{d-1}$-cofactor matroid from approximation theory, Whiteley made ananalogous conjecture in 1996 that the generic $C_{d-2}^{d-1}$-cofactor matroidis the unique maximal abstract $d$-rigidity matroid for all $d\geq 2$. Weverify the case $d=3$ of Whiteley's conjecture in this paper. A key step in ourproof is to verify a second conjecture of Whiteley that the `doubleV-replacement operation' preserves independence in the generic $C_2^1$-cofactormatroid.

Abstract 3-Rigidity and Bivariate $C_2^1$-Splines I: Whiteley's Maximality Conjecture