  The question of the exact region in the complex plane of the possible single
eigenvalues of all $n$-by-$n$ stochastic matrices was raised by Kolmogorov in
1937 and settled by Karpelevi\v{c} in 1951 after a partial result by Dmitriev
and Dynkin in 1946. The Karpelevi\v{c} result is unwieldy, but a simplification
was given by {\DJ}okovi\'c in 1990 and Ito in 1997. The Karpelevi\v{c} region
is determined by a set of boundary arcs each connecting consecutive roots of
unity of order less than $n$. It is shown here that each of these arcs is
realized by a single, somewhat simple, parametrized stochastic matrix. Other
observations are made about the nature of the arcs and several further
questions are raised. The doubly stochastic analog of the Karpelevi\v{c} region
remains open, but a conjecture about it is amplified.
