A simpler construction of volume polynomials for a polyhedron
This article builds on earlier work to give a simple constructive proof of the existence of so-called ``volume polynomials'' for (triangulated) spherical polyhedra. For a polyhedron with a given combinatorial structure, a volume polynomial is a polynomial in \(V^2\) whose coefficients are fixed functions of the edge-lengths of the polyhedron, which has as a root the volume \(V\) of the polyhedron. An example is Franscesca's formula \[ \begin{multlined} 144V^2=-abc-ab'c'-a'bc'-a'b'c+(a+a'+b+b'-c-c')\\ cc'+(a+a'-b-b'+c+c')bb'+(-a- a'+b+b'+c+c')aa'\end{multlined} \] for the volume of the tetrahedron where \((a,a')\), \((b,b')\) and \((c,c')\) are pairs of squares of lengths of opposite edges. It was shown by \textit{I. Kh. Sabitov} [Fundam. Prikl. Mat. 2, 1235--1246 (1996; Zbl 0904.52002), Discrete Comput. Geom. 20, 405--425 (1998; Zbl 0922.52006)], that volume polynomials exist for all spherical triangulations. In the article under review, a simplification of the earlier proof is given, by proving and then using the fact that every triangulation of the 2-sphere contains a vertex which is not incident with any non-facial 3-cycle of edges. The final section of the article examines the case of triangulated toroids and shows under what circumstances they may be cut to yield triangulated spheres with properties the earlier sections show to be important in the search for low-degree volume polynomials.
scientific article; zbMATH DE number 1784607