  We compute the dimension of the center of the 0-Hecke algebra $\mathcal{H}_n$
and of the Nilcoxeter algebra $\mathcal{NC}_n$ using a calculus of diagrams on
the M\"{o}bius band. In the case of the Nilcoxeter algebra, this calculus is
shown to produce a basis for $Z(\mathcal{NC}_n)$ and the table of
multiplication in this basis is shown to be trivial. We conjecture that a basis
for $Z(\mathcal{H}_n)$ can also be obtained in a specific way from this
topological calculus.
