Virtual Knot Theory

TL;DR

This paper introduces virtual knot theory, a generalization of classical knot theory that studies non-planar Gauss codes using virtual crossings, and explores its fundamental properties, invariants, and applications.

Contribution

It provides foundational results, examples, and extensions of invariants in virtual knot theory, expanding the understanding of non-planar knots and their properties.

Findings

Existence of non-trivial virtual knots with trivial Jones polynomial

Development of fundamental group and quandle invariants for virtual knots

Extensions of polynomial and quantum invariants to virtual knots

Abstract

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss codes via knot diagrams with virtual crossings. This paper gives basic results and examples (such as non-trivial virtual knots with trivial Jones polynomial), studies fundamental group and quandles of virtual knots, extensions of the bracket and Jones polynomials, quantum link invariants with virtual framings, Vassiliev invariants and applications to knots in thickened surfaces.