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

TL;DR

This paper proves Whiteley's conjecture for the case d=3, demonstrating that the generic C_{1}^{2}-cofactor matroid is the unique maximal abstract 3-rigidity matroid, by verifying a key operation preserves independence.

Contribution

The paper confirms Whiteley's conjecture for d=3, establishing the maximality of the generic C_{1}^{2}-cofactor matroid in 3-dimensional rigidity theory.

Findings

Verification of Whiteley's conjecture for d=3

Proof that the double V-replacement operation preserves independence

Establishment of the uniqueness of the maximal abstract 3-rigidity matroid

Abstract

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 $d\geq 2$. 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.