Orderable and Deformable Coxeter Hyperbolic Tetrahedrons with Finite Volume

In this article, we classified the 3-dimensional orderable and deformable Coxeter hyperbolic tetrahedrons with finite volume. Andreev’s states a necessary and sufficient condition for a hyperbolic polyhedron with finite volume. Choi’s theorem provides a necessary and sufficient condition for orderable and deformable Coxeter polyhedron. Using graph theory and combinatorics, we find that there are 13 orderable and deformable Coxeter hyperbolic tetrahedrons with finite volume up to symmetry.