  The equilibrium statistical mechanics of classical directed polymers in 2
dimensions is well known to be equivalent to the imaginary-time quantum
dynamics of a 1+1-dimensional many-particle system, with polymer configurations
corresponding to particle world-lines. This equivalence motivates the
application of techniques originally designed for one-dimensional many-particle
quantum systems to the exploration of many-polymer systems, as first recognized
and exploited by P.-G. de Gennes [J.\ Chem.\ Phys.\ {\bf 48}, 2257 (1968)]. In
this low-dimensional setting interactions give rise to an emergent polymer
fluid, and we examine how topological constraints on this polymer fluid (e.g.,
due to uncrossable pins or barriers) and their geometry give rise to strong,
entropy-driven forces. In the limit of large polymer densities, in which a type
of mean-field theory is accurate, we find that a point-like pin causes a
divergent pile-up of polymer density on the high-density side of the pin and a
zero-density region (or gap) of finite area on the low-density. In addition, we
find that the force acting on a pin that is only mildly displaced from its
equilibrium position is sub-Hookean, growing less than linearly with the
displacement, and that the gap created by the pin also grows sublinearly with
the displacement. By contrast, the forces acting between multiple pins
separated along the direction preferred by the polymers are super-Hookean.
These nonlinear responses result from effective long-ranged interactions
between polymer segments, which emerge via short-ranged interactions between
distant segments of long polymer strands. In the present paper, we focus on the
case of an infinitely strong, repulsive contact interaction, which ensures that
the polymers completely avoid one another. In a companion paper, we consider
the effects of a wider set of inter-polymer interactions.
