ÌìÃÀÂ鶹

In simplicial decomposition, we define two invariants --- V_Z and V_Q --- which represent notions of integral and rational volume of a certain class of simplicial complexes. We prove V_Z and V_Q are additive under disjoint union and connected sum, and investigate `integrality gaps' between the two quantities. We apply the theory to establish a conjecture of Sleator, Thurston, and Tarjan on tetrahedral fillings, and, as a corollary, obtain a new proof of Pournin's 2012 result on the diameter of the associahedron. 

 

In simplicial realization, we provide practical sufficient conditions and computer code to prove the existence of Euclidean embeddings of simplicial complexes with specified edge lengths. Example applications include proving the existence of specified edge-length graph embeddings in the plane, or specified edge-length embeddings of polyhedra in space. We apply this work to prove that every triangulation of the 2-sphere with maximum degree <=6 and at most 23 vertices may be embedded in Euclidean 3-space with unit length edges.