Classical And Virtual Pseudodiagram Theory And New Bounds On Unknotting Numbers And Genus
A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we analyze the trivializing number of a pseudodiagram, i.e. the minimum number of crossings that must be resolved to produce the unknot. We consider how much crossing information is needed in a virtual pseudodiagram 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.