Classical and Virtual Pseudodiagram Theory and New Bounds on Unknotting Numbers and Genus

TL;DR

This paper extends pseudodiagram theory to virtual knots, providing new bounds on unknotting numbers and genus, and explores how crossing information determines knot triviality or non-triviality.

Contribution

It generalizes pseudodiagram theory to virtual knots and introduces new bounds on unknotting number, virtual unknotting number, and genus.

Findings

Extended pseudodiagram theory to virtual knots

Derived new upper bounds on unknotting and genus

Analyzed crossing information needed for knot identification

Abstract

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.