Introduction to Virtual Knot Theory

TL;DR

This paper introduces virtual knot theory, presenting new polynomial invariants, categorifications, and parity-based methods, with examples demonstrating their effectiveness in distinguishing virtual knots.

Contribution

It provides a comprehensive exposition of virtual knot invariants, including the parity bracket polynomial, arrow polynomial, and their categorifications, along with new examples and counterexamples.

Findings

Parity methods can distinguish knots not separated by Khovanov homology.

Categorification of the arrow polynomial offers finer distinctions between virtual knots.

Examples show the effectiveness of new invariants in virtual knot classification.

Abstract

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory both combinatorially and in terms of the knot theory in thickened surfaces. The arrow polynomial (of Dye and Kauffman) is a natural generalization of the Jones polynomial, obtained by using the oriented structure of diagrams in the state sum. The paper discusses uses of parity pioneered by Vassily Manturov and uses his parity bracket polynomial to give a counterexample to a conjecture of Fenn, Kauffman and Manturov. The paper gives an exposition of the categorification of the arrow polynomial due to Dye, Kauffman and Manturov and it gives one example (from many found by Aaron Kaestner) of a pair of virtual knots that are not distinguished by Khovanov homology (mod 2), or by the arrow polynomial, but are distinguished by a categorification of the arrow polynomial. Other examples of parity calculations are indicated. For example, this same pair is distinguished by the parity bracket polynomial.