  The work in this article is inspired by a classical problem: the statistical
physical properties of a closed polymer loop that is wound around a rod.
Historically the preserved topology of this system has been addressed through
identification of similarities with magnetic systems. We treat the topological
invariance in terms of a set of rules that describe all augmentations by
additional arcs of some fundamental basic loop of a given winding number. These
augmentations satisfy the Reidemeister move relevant for the polymer with
respect to the rod. The topologically constrained polymer partition function is
now constructed using the combinatorics of allowed arc additions and their
appropriate statistical weights. We illustrate how, for winding number 1, we
can formally derive expressions for lower and upper bounds on the partition
function. Using the lower bound approximation we investigate a flexible polymer
loop wound between two slits, calculating the force on the slit as well as the
average numbers of arc types in dependence of slit width and separation.
Results may be extended to higher winding numbers. The intuitive nature of this
combinatoric scheme allows the development of a variety of approximations and
generalisations.
