Constructing pathological symmetric abstract rigidity matroids

Abstract: We prove that for each $d \geq 2$, there exist infinitely many symmetric abstract $d$-rigidity matroids that are not the $d$-rigidity matroid, the $d$-hyperconnectivity matroid, or the $d$-cofactor matroid. When $d=2$, our constructed matroids are pathological, in the sense that they are not contained within any abstract 2-rigidity matroid family. Our construction additionally provides a counter-example to a conjecture of Tyomkyn.