Linking Rigid Bodies Symmetrically Bernd Schulze 1 and Shin-ichi - - PowerPoint PPT Presentation

Linking Rigid Bodies Symmetrically

Bernd Schulze1 and Shin-ichi Tanigawa2

1Lancaster Unviersity, 2Kyoto 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 → Rd

◮ (G, p) is flexible if ∃ a continuos ”deformation” keeping the edge

lengths; otherwise (G, p) is rigid

2 / 14

Infinitesimal Rigidity

◮ ˙

p : V → Rd 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 ∈ Rd such that ˙ p(i) = Sp(i) + t for i ∈ V .

3 / 14

Infinitesimal Rigidity

◮ ˙

p : V → Rd 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 ∈ Rd 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 → Rd 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 ∈ Rd 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 R2 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 → Rd 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 ∈ Rd 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 R2 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 R3 if and only if 5G contains six edge-disjoint spanning trees. G G 2 5G

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

Rd, 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

Rd, for each e ∈ E.

◮ Theorem (Tay 89, Whiteley 88). Suppose h is generic. Then (G, h)

is infinitesimally rigid in Rd if and only if ( d+1

2

• − 1)G contains

d+1

2

• edge-disjoint spanning trees.

6 / 14

Molecular Frameworks as Body-hinge Frameworks

◮ In (G 2, p), NG(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 G 2

7 / 14

Molecular Frameworks as Body-hinge Frameworks

◮ In (G 2, p), NG(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 Rd if and only if ( d+1

2

• − 1)G

contains d+1

2

• edge-disjoint spanning trees.

G G 2

7 / 14

Molecular Frameworks as Body-hinge Frameworks

◮ In (G 2, p), NG(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 Rd if and only if ( d+1

2

• − 1)G

contains d+1

2

• edge-disjoint spanning trees.

◮ Here we give a symmetric version of Tay-Whiteley’s theorem for

body-hinge frameworks. G G 2

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(Rd)

8 / 14

Quotient Signed Graphs

◮ Definition For a Z2-symmetric graph G, the quotient signed graph is

a pair (G/Z2, 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

− − + + + + + +

(G/Z2, ψ)

− + + +

9 / 14

Combinatorial Characterization for Cs in R3

◮ Cs: a group generated by a reflection in R3 ◮ (G, h): a (Z2, θ, τ)-symmetric body-hinge framework, where

◮ θ : Z2 → Aut(G), freely acting on E(G) ◮ τ : Z2 → Cs, faithful ◮ h: a Cs-generic hinge-configuration

◮ Theorem(Schulze&T14) (G, h) is infinitesimally rigid if and only if

the quotient signed graph (5G/Z2, ψ) contains edge-disjoint

◮ three spanning trees and ◮ three spanning unbalanced 1-forests

− − + + + + + +

10 / 14

Combinatorial Characterization for C2 in R3

◮ C2: a group generated by a rotation in R3 ◮ (G, h): a (Z2, θ, ρ)-symmetric body-hinge framework, where

◮ θ : Z2 → Aut(G), freely acting on E(G) ◮ τ : Z2 → C2, faithful ◮ h: a C2-generic hinge-configuration

◮ Theorem(Schulze&T14) (G, h) is infinitesimally rigid if and only if

the quotient signed graph (5G/Z2, 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 Rd isomorphic to Γ, ◮ τ : Γ → P, an isomorphism ◮ ˆ

τ : γ ∈ Γ → τ(γ) 1

• ∈ O(Rd+1)

◮ C2(ˆ

τ) : γ ∈ Γ → C2(ˆ τ(γ)) ∈ O(2 Rd+1),

◮ For a (d + 1) × (d + 1)-matrix A, C2(A) denotes the second

compound matrix of A; that is, a matrix of size d+1

2

• ×

d+1

2

• formed

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 = (Z2)k ◮ P: a point group of Rd with an isomorphism τ : Γ → P ◮ C2(ˆ

τ) =

1≤i≤ d+1

2

τi, where τi : Γ → {−, +} ◮ For 1 ≤ j ≤ 2k, ρj : Γ → {−, +}: irreducible representations of Γ.

13 / 14

◮ Suppose Γ = (Z2)k ◮ P: a point group of Rd with an isomorphism τ : Γ → P ◮ C2(ˆ

τ) =

1≤i≤ d+1

2

τi, where τi : Γ → {−, +} ◮ For 1 ≤ j ≤ 2k, ρj : Γ → {−, +}: irreducible representations of Γ. ◮ G: a (Γ, θ)-symmetric graph ◮ (G/Γ, ψ): the quotient Γ-labeled graph

13 / 14

◮ Suppose Γ = (Z2)k ◮ P: a point group of Rd with an isomorphism τ : Γ → P ◮ C2(ˆ

τ) =

1≤i≤ d+1

2

τi, where τi : Γ → {−, +} ◮ For 1 ≤ j ≤ 2k, ρj : Γ → {−, +}: irreducible representations of Γ. ◮ G: a (Γ, θ)-symmetric graph ◮ (G/Γ, ψ): the quotient Γ-labeled graph ◮ For 1 ≤ i ≤

d+1

2

• and 1 ≤ j ≤ 2k, define ψi,j : E(G/Γ) → {−, +}

by ψi,j : e → ρj(ψ(e)) · τi(ψ(e))

13 / 14

◮ Suppose Γ = (Z2)k ◮ P: a point group of Rd with an isomorphism τ : Γ → P ◮ C2(ˆ

τ) =

1≤i≤ d+1

2

τi, where τi : Γ → {−, +} ◮ For 1 ≤ j ≤ 2k, ρj : Γ → {−, +}: irreducible representations of Γ. ◮ G: a (Γ, θ)-symmetric graph ◮ (G/Γ, ψ): the quotient Γ-labeled graph ◮ For 1 ≤ i ≤

d+1

2

• and 1 ≤ j ≤ 2k, define ψi,j : E(G/Γ) → {−, +}

by ψi,j : e → ρj(ψ(e)) · τi(ψ(e))

13 / 14

◮ Suppose Γ = (Z2)k ◮ P: a point group of Rd with an isomorphism τ : Γ → P ◮ C2(ˆ

τ) =

1≤i≤ d+1

2

τi, where τi : Γ → {−, +} ◮ For 1 ≤ j ≤ 2k, ρj : Γ → {−, +}: irreducible representations of Γ. ◮ G: a (Γ, θ)-symmetric graph ◮ (G/Γ, ψ): the quotient Γ-labeled graph ◮ For 1 ≤ i ≤

d+1

2

• and 1 ≤ j ≤ 2k, define ψi,j : E(G/Γ) → {−, +}

by ψi,j : e → ρj(ψ(e)) · τi(ψ(e))

◮ Theorem (Schulze&T14). Let (G, h) be a (Γ, θ, τ)-symmetric

body-hinge framework with a P-generic hinge-configuration h. Then (G, h) is infinitesimally rigid iff, for each 1 ≤ j ≤ 2k, ( d+1

2

• − 1)G/Γ

contains a spanning subgraph Hj such that

◮ |E(Hj)| =

d+1

2

• |V (Hj)| −

1 |Γ|

• γ∈Γ Trace(ρj(γ)ˆ

τ (2)(γ))

◮ Hj contains

d+1

2

• edge-disjoint subgraphs Hi

j (1 ≤ i ≤

d+1

2

• ) such

that each Hi

j is a spanning unbalanced 1-forest with respect to ψi,j.

13 / 14

Concluding Remarks

◮ Extension to molecular frameworks (hinge-identified body-hinge

frameworks)?

◮ maybe possible for Cs or C2

◮ Extension to a wider class of point groups?

◮ Our proof uses Whiteley’s idea, which requires a tree-decomposition

property of graphs satisfying a necessary ”count” condition

◮ Applications to protein-function analysis?

14 / 14