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arXiv · 1911.00207

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

Katie Clinch·Bill Jackson·Shin-ichi Tanigawa
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Abstract

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 independence in this matroid. This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks. We use our characterization to verify that the counterparts of conjectures of Dress (on the rank function) and Lov\'{a}sz and Yemini (which suggested a sufficient connectivity condition for rigidity) hold for this matroid.

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Katie Clinch, Bill Jackson, Shin-ichi Tanigawa. 2019-11-01. Abstract 3-Rigidity and Bivariate $C_2^1$-Splines II: Combinatorial Characterization. https://doi.org/10.19086/da.34692

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