Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 1, pp. 261-273 (2002)

A Simpler Construction of Volume Polynomials for a Polyhedron

Serge Lawrencenko, Seiya Negami, Idjad Kh. Sabitov

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA, e-mail:
Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-Ku, Yokohama 240-8501, Japan, e-mail:
Faculty of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia, e-mail:

Abstract: This paper is related to the third author's previous result on the existence of volume polynomials for a given polyhedron having only triangular faces. We simplify his original proof in the case when the polyhedron is homeomorphic to the $2$-sphere. Our approach exploits the fact that any such polyhedron contains a so-called clean vertex - that is, a vertex not incident with any nonfacial cycle composed of $3$ edges. This fact appears as one of the main results of the article. Also, we characterize triangulations reducible to a tetrahedron by repeatedly removing $3$-valent vertices, and estimate the degree of volume polynomials. We address the torus case too.

Keywords: $2$-sphere, torus, triangulation, polyhedron, volume

Classification (MSC2000): 52B05; 51M25, 57M15, 57Q15

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