  We probe the character of knotting in open, confined polymers, assigning knot
types to open curves by identifying their projections as virtual knots. In this
sense, virtual knots are transitional, lying in between classical knot types,
which are useful to classify the ambiguous nature of knotting in open curves.
Modelling confined polymers using both lattice walks and ideal chains, we find
an ensemble of random, tangled open curves whose knotting is not dominated by
any single knot type, a behaviour we call weakly knotted. We compare cubically
confined lattice walks and spherically confined ideal chains, finding the weak
knotting probability in both families is quite similar and growing with length,
despite the overall knotting probability being quite different. In contrast,
the probability of weak knotting in unconfined walks is small at all lengths
investigated. For spherically confined ideal chains, weak knotting is strongly
correlated with the degree of confinement but is almost entirely independent of
length. For ideal chains confined to tubes and slits, weak knotting is
correlated with an adjusted degree of confinement, again with length having
negligible effect.
