  We consider the three-state toric homogeneous Markov chain model (THMC)
without loops and initial parameters. At time $T$, the size of the design
matrix is $6 \times 3\cdot 2^{T-1}$ and the convex hull of its columns is the
model polytope. We study the behavior of this polytope for $T\geq 3$ and we
show that it is defined by 24 facets for all $T\ge 5$. Moreover, we give a
complete description of these facets. From this, we deduce that the toric ideal
associated with the design matrix is generated by binomials of degree at most
6. Our proof is based on a result due to Sturmfels, who gave a bound on the
degree of the generators of a toric ideal, provided the normality of the
corresponding toric variety. In our setting, we established the normality of
the toric variety associated to the THMC model by studying the geometric
properties of the model polytope.
