  Potts spin systems play a fundamental role in statistical mechanics and
quantum field theory, and can be studied within the spin, the Fortuin-Kasteleyn
(FK) bond or the $q$-flow (loop) representation. We introduce a Loop-Cluster
(LC) joint model of bond-occupation variables interacting with $q$-flow
variables, and formulate a LC algorithm that is found to be in the same
dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to
a theoretical unification for all the representations, and numerically, one can
apply the most efficient algorithm in one representation and measure physical
quantities in others. Moreover, by using the LC scheme, we construct a
hierarchy of geometric objects that contain as special cases the $q$-flow
clusters and the backbone of FK clusters, the exact values of whose fractal
dimensions in two dimensions remain as an open question. Our work not only
provides a unified framework and an efficient algorithm for the Potts model,
but also brings new insights into rich geometric structures of the FK clusters.
