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

@article{Clinch2019Abstract3A, title={Abstract 3-Rigidity and Bivariate \$C\_2^1\$-Splines I: Whiteley's Maximality Conjecture}, author={Katie Clinch and Bill Jackson and Shin-ichi Tanigawa}, journal={arXiv: Combinatorics}, year={2019} }

A long-standing conjecture in rigidity theory 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 3-dimensional rigidity matroid and the generic $C_2^1$-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic $C_2^1$-cofactor matroid is the unique maximal abstract 3-rigidity matroid. We verify…

## 3 Citations

### Bar-and-joint rigidity on the moment curve coincides with cofactor rigidity on a conic

- Mathematics
- 2021

. We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the C d − 2 d − 1 -cofactor rigidity of…

### Maximal Matroids in Weak Order Posets

- Mathematics
- 2021

Let X be a family of subsets of a finite set E. A matroid on E is called an X matroid if each set in X is a circuit. We consider the problem of determining when there exists a unique maximal X…

### Abstract 3-Rigidity and Bivariate $C_2^1$-Splines II: Combinatorial Characterization.

- Mathematics
- 2019

We showed in the first paper of this series that the generic $C_2^1$-cofactor matroid is the unique maximal abstract $3$-rigidity matroid. In this paper we obtain a combinatorial characterization of…

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