Abstract
We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all ℓp spaces with p≠2). Complete combinatorial characterisations are obtained for half-turn rotation in the ℓ1 and ℓ∞-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight gain graphs.
Highlights
Xχ andBy Lemma 3.5, Hab and Hac are connected and so it follows that Hab ∪ Hac contains an unbalanced cycle
We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space
For a large class of d-dimensional normed spaces, with d ≥ 2, this leads to the identification of (d, d, m)-gain-tight gain graphs, with m ∈ {0, 1, 2, d − 2}, as the underlying structure graphs for phase-symmetrically isostatic frameworks with rotational symmetry
Summary
By Lemma 3.5, Hab and Hac are connected and so it follows that Hab ∪ Hac contains an unbalanced cycle This is a contradiction and so v is admissible since adding the edge (ac, −1) will not violate (2, 2, 0)-gain-sparsity. Consider a reduction at v adding a loop at a and the edge (bc, βγ) If this move is not admissible there is a balanced subgraph Hbc of G0 −v containing b, c with f (Hbc) = 2 in which all paths from b to c have gain βγ. We let K4 be the (2, 2, 0)gain-tight gain graph consisting of a balanced K4 as well as one additional vertex x and the four edges (xa, 1), (xa, −1), (xb, 1), (xb, −1) where a and b are distinct vertices of the K4. We suppose that every Wi is incident to two parallel edges in G∗0
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