  The isospin chemical potential region is known as the sign-problem free
region of quantum chromodynamics (QCD). In this paper, we introduce the isospin
chemical potential to the three-dimensional three-state Potts model to mimic
the dense QCD; e.g., the QCD effective model with heavy quarks at finite
density. We call it as QCD-like Potts model. The QCD-like Potts model does not
have the sign problem, but we can expect that it shares some properties with
QCD. Since we can obtain the non-approximated Potts spin configuration at
finite isospin chemical potential where the simple Metropolis algorithm can
work, we perform the persistent homology analysis towards exploring the dense
spatial structure of QCD. We show that the averaged birth-death ratio has the
same information with the Polyakov loop, but the maximum birth-death ratio has
additional information near the phase transition.
