  The most distant known trans-Neptunian objects (perihelion distance above 38
au and semimajor axis above 150 au) are of interest for their potential to
reveal past, external, or present but unseen perturbers. Realizing this
potential requires understanding how the known planets influence their orbital
dynamics. We use a recently-developed Poincare mapping approach for orbital
phase space studies of the circular planar restricted three body problem, which
we have extended to the case of the three-dimensional restricted problem with
$N$ planetary perturbers. With this approach, we explore the dynamical
landscape of the 23 most distant TNOs under the perturbations of the known
giant planets. We find that, counter to common expectations, almost none of
these TNOs are far removed from Neptune's resonances. Nearly half (11) of these
TNOs have orbits consistent with stable libration in Neptune's resonances; in
particular, the orbits of TNOs 148209 and 474640 overlap with Neptune's 20:1
and 36:1 resonances, respectively. Five objects can be ruled currently
non-resonant, despite their large orbital uncertainties, because our mapping
approach determines the resonance boundaries in angular phase space in addition
to semimajor axis. Only three objects are in orbital regions not appreciably
affected by resonances: Sedna, 2012 VP113 and 2015 KG163. Our analysis also
demonstrates that Neptune's resonances impart a modest (few percent)
non-uniformity in the longitude of perihelion distribution of the currently
observable distant TNOs. While not large enough to explain the observed
clustering, this small dynamical sculpting of the perihelion longitudes could
become relevant for future, larger TNO datasets.
