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

  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},
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… 
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On Abstract Rigidity Matroids

  • V. Nguyen
  • Mathematics
    SIAM J. Discret. Math.
  • 2010
This paper provides a combinatorial characterization of abstract rigidity matroids in any dimension and shows that in dimension 3, however, there exists a 1-extendable abstract rigsidity matroid that is not a generic rigidityMatroid.