Symmetry-forced rigidity of frameworks on surfaces

Abstract

A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordán et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space-as well as for other types of surfaces-we provide some observations and conjectures.

Pairing Symmetries for Euclidean and Spherical Frameworks

“…As in Corollary 3.4 we may deduce the following. Note that (b) was already used in [20]. As for incidental symmetry, for the reflection group C s in R d , we also obtain the following complete analogue of Theorem 1.1, whose proof is similar to Corollary 3.6.…”

“…In [20], combinatorial characterisations for Γ-regular forced Γ-symmetric rigidity on S 2 (where the action θ : Γ → Aut(G) is free on the vertex set) have been established for the groups C s , C n , n ∈ N, C i , C nv , n odd, C nh , n odd, and S 2n , n even. (For the groups C s and C n , for example, the characterisation is the same as the one given in Theorem 2.1 for bar-joint frameworks in R 2 .)…”

“…For the remaining groups, this problem is still open. (See Table 1 in [20] for further details.) The infinitesimal rigidity for incidentally symmetric frameworks on S 2 has not yet been investigated.…”

Count Matroids of Group-Labeled Graphs

“…Our work is in fact motivated from characterizations of the rigidity of graphs with symmetry. Recent works on this subject reveal connections of the infinitesimal rigidity of symmetric bar-joint frameworks with count conditions of the form (2) on the quotient group-labeled graphs [9,10,12,15,7,11], where each symmetry and each rigidity model gives a distinct α ψ . In Section 2 we give examples, several of which were not known to form matroids before.…”

“…In Section 2 we give examples, several of which were not known to form matroids before. In this context it is crucial to know whether a necessary count condition forms a matroid or not (see, e.g., [9,10,15,7,11]). Our construction uses more refined properties of group-labelings than balancedness.…”