Linking Rigid Bodies Symmetrically Bernd Schulze 1 and Shin-ichi Tanigawa 2 1 Lancaster Unviersity, 2 Kyoto University June 8, 2014 1 / 14
Rigidity of Frameworks ◮ A bar-joint framework is a pair ( G , p ) of a simple graph G = ( V , E ) and p : V → R d ◮ ( G , p ) is flexible if ∃ a continuos ”deformation” keeping the edge lengths; otherwise ( G , p ) is rigid 2 / 14
Infinitesimal Rigidity p : V → R d is an infinitesimal motion of ( G , p ) if ◮ ˙ � p ( i ) − p ( j ) , ˙ p ( i ) − ˙ p ( j ) � = 0 ( ∀ ij ∈ E ) . ◮ ( G , p ) is infinitesimally rigid if every infinitesimal motion ˙ p of ( G , p ) is trivial, i.e., ∃ a skew symmetric matrix S and t ∈ R d such that p ( i ) = Sp ( i ) + t for i ∈ V . ˙ 3 / 14
Infinitesimal Rigidity p : V → R d is an infinitesimal motion of ( G , p ) if ◮ ˙ � p ( i ) − p ( j ) , ˙ p ( i ) − ˙ p ( j ) � = 0 ( ∀ ij ∈ E ) . ◮ ( G , p ) is infinitesimally rigid if every infinitesimal motion ˙ p of ( G , p ) is trivial, i.e., ∃ a skew symmetric matrix S and t ∈ R d such that p ( i ) = Sp ( i ) + t for i ∈ V . ˙ ◮ Proposition (Asimov and Roth 79) Suppose p is generic. Then ( G , p ) is rigid iff ( G , p ) is infinitesimally rigid. 3 / 14
Infinitesimal Rigidity p : V → R d is an infinitesimal motion of ( G , p ) if ◮ ˙ � p ( i ) − p ( j ) , ˙ p ( i ) − ˙ p ( j ) � = 0 ( ∀ ij ∈ E ) . ◮ ( G , p ) is infinitesimally rigid if every infinitesimal motion ˙ p of ( G , p ) is trivial, i.e., ∃ a skew symmetric matrix S and t ∈ R d such that p ( i ) = Sp ( i ) + t for i ∈ V . ˙ ◮ Proposition (Asimov and Roth 79) Suppose p is generic. Then ( G , p ) is rigid iff ( G , p ) is infinitesimally rigid. ◮ Theorem (Laman 1970) Suppose p is generic. Then ( G , p ) is minimally rigid in R 2 if and only if ◮ | E | = 2 | V | − 3 and ◮ | E ( G ′ ) | ≤ 2 | V ( G ′ ) | − 3 for any G ′ ⊆ G with | E ( G ′ ) | ≥ 2. 3 / 14
Infinitesimal Rigidity p : V → R d is an infinitesimal motion of ( G , p ) if ◮ ˙ � p ( i ) − p ( j ) , ˙ p ( i ) − ˙ p ( j ) � = 0 ( ∀ ij ∈ E ) . ◮ ( G , p ) is infinitesimally rigid if every infinitesimal motion ˙ p of ( G , p ) is trivial, i.e., ∃ a skew symmetric matrix S and t ∈ R d such that p ( i ) = Sp ( i ) + t for i ∈ V . ˙ ◮ Proposition (Asimov and Roth 79) Suppose p is generic. Then ( G , p ) is rigid iff ( G , p ) is infinitesimally rigid. ◮ Theorem (Laman 1970) Suppose p is generic. Then ( G , p ) is minimally rigid in R 2 if and only if ◮ | E | = 2 | V | − 3 and ◮ | E ( G ′ ) | ≤ 2 | V ( G ′ ) | − 3 for any G ′ ⊆ G with | E ( G ′ ) | ≥ 2. ◮ It is still open to give a 3-dimensional counterpart of Laman’s theorem 3 / 14
Molecular Frameworks ◮ A molecular framework is a bar-joint framework whose underlying graph is G 2 of some G . ◮ Theorem (Katoh&T11) Suppose p is generic. Then ( G 2 , p ) is rigid in R 3 if and only if 5 G contains six edge-disjoint spanning trees. G 2 G 5 G 4 / 14
Symmetric Frameworks Symmetry in proteins... Rigidity of symmetric frameworks ◮ Symmetry-forced rigidity (asking symmetry-preserving motions) ◮ well understood ◮ Infinitesimal rigidity ◮ Rigidity 5 / 14
Body-hinge Frameworks ◮ A body-hinge framework is a structure consisting of rigid bodies connected by hinges... ◮ A body-hinge framework is a pair ( G = ( V , E ) , h ); ◮ vertex ⇔ body ◮ edge ⇔ hinge ◮ h ( e ) := { h ( e ) 1 , . . . , h ( e ) d − 1 } , affinely independent d − 1 points in R d , for each e ∈ E . 6 / 14
Body-hinge Frameworks ◮ A body-hinge framework is a structure consisting of rigid bodies connected by hinges... ◮ A body-hinge framework is a pair ( G = ( V , E ) , h ); ◮ vertex ⇔ body ◮ edge ⇔ hinge ◮ h ( e ) := { h ( e ) 1 , . . . , h ( e ) d − 1 } , affinely independent d − 1 points in R d , for each e ∈ E . ◮ Theorem (Tay 89, Whiteley 88). Suppose h is generic. Then ( G , h ) is infinitesimally rigid in R d if and only if ( � d +1 � − 1) G contains 2 � d +1 � edge-disjoint spanning trees. 2 6 / 14
Molecular Frameworks as Body-hinge Frameworks ◮ In ( G 2 , p ), N G ( v ) ∪ { v } forms a clique, which is rigid ◮ ( G 2 , p ) can be regarded as a hinge-concurrent body-hinge framework ( G , h ) ◮ h is not generic; for each v , span ( h ( e )) intersects at p ( v ) for every e incident to v in G G 2 G 7 / 14
Molecular Frameworks as Body-hinge Frameworks ◮ In ( G 2 , p ), N G ( v ) ∪ { v } forms a clique, which is rigid ◮ ( G 2 , p ) can be regarded as a hinge-concurrent body-hinge framework ( G , h ) ◮ h is not generic; for each v , span ( h ( e )) intersects at p ( v ) for every e incident to v in G ◮ Theorem (Katoh&T11). Suppose ( G , h ) is hinge-concurrent generic. Then ( G , h ) is infinitesimally rigid in R d if and only if ( � d +1 � − 1) G 2 � d +1 � contains edge-disjoint spanning trees. 2 G 2 G 7 / 14
Molecular Frameworks as Body-hinge Frameworks ◮ In ( G 2 , p ), N G ( v ) ∪ { v } forms a clique, which is rigid ◮ ( G 2 , p ) can be regarded as a hinge-concurrent body-hinge framework ( G , h ) ◮ h is not generic; for each v , span ( h ( e )) intersects at p ( v ) for every e incident to v in G ◮ Theorem (Katoh&T11). Suppose ( G , h ) is hinge-concurrent generic. Then ( G , h ) is infinitesimally rigid in R d if and only if ( � d +1 � − 1) G 2 � d +1 � contains edge-disjoint spanning trees. 2 ◮ Here we give a symmetric version of Tay-Whiteley’s theorem for body-hinge frameworks. G 2 G 7 / 14
Symmetric Body-hinge Frameworks ◮ A graph G = ( V , E ) is (Γ , θ )-symmetric (or, simply, Γ-symmetric) if Γ is isomorphic to a subgroup of Aut ( G ) through θ : Γ → Aut ( G ). ◮ A body-hinge framework ( G , h ) is (Γ , θ, τ )-symmetric (or, simply, Γ-symmetric) if τ ( γ ) h ( e ) i = h ( θ ( γ ) e ) i ( ∀ e ∈ E , ∀ i ∈ { 1 , . . . , d − 1 } ) where τ : Γ → O ( R d ) 8 / 14
Quotient Signed Graphs ◮ Definition For a Z 2 -symmetric graph G , the quotient signed graph is a pair ( G / Z 2 , ψ ) of the quotient graph G / Γ and ψ : E ( G ) → {− , + } . ◮ Definition A cycle is negative if it contains an odd number of negative edges ◮ Definition A signed graph is called an unbalanced 1-forest if each connected component contains exactly one cycle, which is negative. − + + + + + + + + + − − ( G / Z 2 , ψ ) G 9 / 14
Combinatorial Characterization for C s in R 3 ◮ C s : a group generated by a reflection in R 3 ◮ ( G , h ): a ( Z 2 , θ, τ )-symmetric body-hinge framework, where ◮ θ : Z 2 → Aut ( G ), freely acting on E ( G ) ◮ τ : Z 2 → C s , faithful ◮ h : a C s -generic hinge-configuration ◮ Theorem(Schulze&T14) ( G , h ) is infinitesimally rigid if and only if the quotient signed graph (5 G / Z 2 , ψ ) contains edge-disjoint ◮ three spanning trees and ◮ three spanning unbalanced 1-forests − + + + + + + − 10 / 14
Combinatorial Characterization for C 2 in R 3 ◮ C 2 : a group generated by a rotation in R 3 ◮ ( G , h ): a ( Z 2 , θ, ρ )-symmetric body-hinge framework, where ◮ θ : Z 2 → Aut ( G ), freely acting on E ( G ) ◮ τ : Z 2 → C 2 , faithful ◮ h : a C 2 -generic hinge-configuration ◮ Theorem(Schulze&T14) ( G , h ) is infinitesimally rigid if and only if the quotient signed graph (5 G / Z 2 , ψ ) contains edge-disjoint ◮ two spanning trees and ◮ four spanning unbalanced 1-forests − + + + + + + − 11 / 14
More generally ◮ Γ: a finite group ◮ P : a point group of R d isomorphic to Γ, ◮ τ : Γ → P , an isomorphism � τ ( γ ) � 0 ◮ ˆ ∈ O ( R d +1 ) τ : γ ∈ Γ �→ 0 1 τ ( γ )) ∈ O ( � 2 R d +1 ), ◮ C 2 (ˆ τ ) : γ ∈ Γ �→ C 2 (ˆ ◮ For a ( d + 1) × ( d + 1)-matrix A , C 2 ( A ) denotes the second � d +1 � d +1 � � compound matrix of A ; that is, a matrix of size × formed 2 2 from all the 2 × 2 minors det A [ { i , j } , { k , l } ] arranged with the index sets { i , j } and { k , l } in lexicographic order. 12 / 14
◮ Suppose Γ = ( Z 2 ) k ◮ P : a point group of R d with an isomorphism τ : Γ → P � τ i , where τ i : Γ → {− , + } ◮ C 2 (ˆ τ ) = � � d +1 1 ≤ i ≤ 2 ◮ For 1 ≤ j ≤ 2 k , ρ j : Γ → {− , + } : irreducible representations of Γ. 13 / 14
◮ Suppose Γ = ( Z 2 ) k ◮ P : a point group of R d with an isomorphism τ : Γ → P � τ i , where τ i : Γ → {− , + } ◮ C 2 (ˆ τ ) = � � d +1 1 ≤ i ≤ 2 ◮ For 1 ≤ j ≤ 2 k , ρ j : Γ → {− , + } : irreducible representations of Γ. ◮ G : a (Γ , θ )-symmetric graph ◮ ( G / Γ , ψ ): the quotient Γ-labeled graph 13 / 14
◮ Suppose Γ = ( Z 2 ) k ◮ P : a point group of R d with an isomorphism τ : Γ → P � τ i , where τ i : Γ → {− , + } ◮ C 2 (ˆ τ ) = � � d +1 1 ≤ i ≤ 2 ◮ For 1 ≤ j ≤ 2 k , ρ j : Γ → {− , + } : irreducible representations of Γ. ◮ G : a (Γ , θ )-symmetric graph ◮ ( G / Γ , ψ ): the quotient Γ-labeled graph � d +1 ◮ For 1 ≤ i ≤ � and 1 ≤ j ≤ 2 k , define ψ i , j : E ( G / Γ) → {− , + } 2 by ψ i , j : e �→ ρ j ( ψ ( e )) · τ i ( ψ ( e )) 13 / 14
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