Abstract 3-Rigidity and Bivariate C-2(1)-Splines I: Whiteley's Maximality Conjecture

A conjecture of Graver from 1991 states that the generic 3-dimensional rigidity matroid is the unique maximal abstract 3-rigidity matroid with respect to the weak order on matroids. Based on a close similarity between the generic d-dimensional rigidity matroid and the generic C-d-2(d-1)-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic C-d-2(d-1)-cofactor matroid is the unique maximal abstract d-rigidity matroid for all d >= 2. We verify the case d = 3 of Whiteley's conjecture in this paper. A key step in our proof is to verify a second conjecture of Whiteley that the 'double V-replacement operation' preserves independence in the generic C-2(1)-cofactor matroid.