  We introduce a framework for adsorption of a single polymer in which the
topology of the polymer is quenched before adsorption, in contrast to more
standard adsorption models having annealed topology. Our "topology" refers
either to the precise branching structure of a branched polymer (in any
dimension), or else to the knot type of a ring polymer in three dimensions. The
quenched topology is chosen uniformly at random from all lattice polymers of a
given size in one of four classes (lattice animals, trees, combs, or rings),
and we then consider adsorption of the subclass of configurations that have the
quenched topology. When the polymer-surface attraction increases without bound,
the quenched topological structure keeps a macroscopic fraction of monomers off
the surface, in contrast with annealed models that asymptotically have 100% of
monomers in the surface. We prove properties of the limiting free energy and
the critical point in each model, although important open questions remain. We
pay special attention to the class of comb polymers, which admit some rigorous
answers to questions that otherwise remain open. Since the class of all combs
was not previously examined rigorously in full generality, we also prove the
existence of its growth constant and its limiting free energy for annealed
adsorption.
