This paper is related to the third author's previous result on the exis- tence 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 com- posed 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. MSC 2000: 52B05 (primary); 51M25, 57M15, 57Q15 (secondary)
[1]
A canonical polynomial for the volume of a polyhedron
,
1999
.
[2]
S. Lawrencenko.
Irreducible triangulations of a torus
,
1987
.
[3]
J. Bokowski,et al.
All Realizations of Möbius' Torus with 7 Vertices
,
1991
.
[4]
Idzhad Kh. Sabitov,et al.
The Volume as a Metric Invariant of Polyhedra
,
1998,
Discret. Comput. Geom..
[5]
Seiya Negami,et al.
Uniqueness and faithfulness of embedding of toroidal graphs
,
1983,
Discret. Math..
[6]
S. A. Lavrenchenko,et al.
Irreducible triangulations of the torus
,
1990
.