  A pseudodiagram is a diagram of a knot with some crossing information
missing. We review and expand the theory of pseudodiagrams introduced by R.
Hanaki. We then extend this theory to the realm of virtual knots, a
generalization of knots. In particular, we investigate how much crossing
information must be known to conclude that a diagram is a diagram of the unknot
(the trivializing number). We also consider how much information is necessary
to identify a non-trivial knot, a classical knot, or a non-classical knot. We
then apply pseudodiagram theory to develop new upper bounds on unknotting
number, virtual unknotting number, and genus.
