  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.

Geometric Topology