[en] Direct numerical simulations (DNS) of the transition to and fully developed elasto-inertial turbulence (EIT) of a polymer solution in a channel flow has been used as a basis for the study of the topology and dynamics of these flows. The Reynolds number in these DNS ranged from 500 to 5000. The topology of these flows was studied through the joint probability density functions (JPDFs) of the second and third invariants of the velocity gradient tensor (VGT), $Q_A$ and $R_A$ respectively and the JPDFs of the second invariants of the rate-of-strain tensor and the rate-of-rotation tensor, $Q_S$ and $Q_W$ respectively. The results suggest that these transitional and fully developed EIT flows are predominantly made up of vortex sheets. Dynamic mode decomposition has been undertaken on the second invariant of the VGT, $Q_A$, which reveals that the most amplified mode is a two-dimensional structure located in the near-wall region. A ``discontinuity'' is observed close to the wall, which corresponds closely to the location of extrema of the mean polymer extension and is hypothesized to be a critical layer.