  The impact of polymer-polymer interactions of various types on the
thermodynamics, structure, and accommodation of topological constraints is
addressed for systems comprising many directed polymers in two spatial
dimensions. The approach is predicated on the well-known equivalence between
the classical equilibrium statistical mechanics of directed polymers in two
spatial dimensions and the imaginary-time quantum dynamics of particles in one
spatial dimension, originally exploited by P.-G. de Gennes [J.\ Chem.\ Phys.\
{\bf 48}, 2257 (1968)]. Known results concerning two exactly solvable
microscopic models of quantum particles moving in one spatial dimension---the
Lieb-Liniger model of contact interactions and the Calogero-Sutherland model of
long-range interactions---are used to shed light on the behavior of the
corresponding polymeric systems. In addition, the technique of bosonization is
used to reveal how generic polymer interactions give rise to an emergent
polymer fluid that has universal collective excitations. Comparison of the
response to topological constraints of a fluid of simply noncrossing (i.e.,
noncrossing but otherwise noninteracting) directed polymers, explored in a
companion Paper, to the response of a generically interacting directed polymer
fluid reveals that the structure is quantitatively unchanged by the generic
interactions on the line transverse to the pin, and is qualitatively unchanged
by the generic interactions throughout the two dimensions of the system's
extent. Furthermore, the free-energy cost associated with a pin that partitions
a system having generic interactions is found to be proportional to the
pin-partitioning cost for a system of simply noncrossing polymers.
