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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