论文标题
三角歧管的全球刚度
Global Rigidity of Triangulated Manifolds
论文作者
论文摘要
我们证明，如果$ g $是连接的三角形$（d-1）$ - 歧管的图形，则对于$ d \ geq 3 $，则$ g $通常在$ \ mathbb r^d $中通常是全球刚性的，并且仅当它是$（d+1）$ - 连接时，如果$ d = $ d = 3 $ d = 3 $，$ g $不是平面。特殊情况$ d = 3 $验证了康纳利的猜想。我们的结果实际上适用于更大类的简单络合物，即简单曲霉的电路。我们还提供了我们主要定理的两个重要应用。我们表明，具有下限定理给出的极端边缘数的伪遵循的表征延伸到简单曲霉的电路。我们还证明了卡莱（Kalai）猜想的通用情况，即从其应力空间中对多型的重构性的重构性。我们的主要结果的证据适应了早期的雾剂和白色的思想，以设置全球刚性的设置。特别是，我们验证了怀特利（Whiteley）的顶点分裂猜想的特殊情况。
We prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case $d=3$ verifies a conjecture of Connelly. Our results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. We also give two significant applications of our main theorems. We show that that the characterisation of pseudomanifolds with extremal edge numbers given by the Lower Bound Theorem extends to circuits of the simplicial matroid. We also prove the generic case of a conjecture of Kalai concerning the reconstructability of a polytope from its space of stresses. The proofs of our main results adapt earlier ideas of Fogelsanger and Whiteley to the setting of global rigidity. In particular we verify a special case of Whiteley's vertex splitting conjecture for global rigidity.
