  Two categorifications are given for the arrow polynomial, an extension of the
Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the
bracket polynomial to infinitely many variables, each variable corresponding to
an integer {\it arrow number} calculated from each loop in an oriented state
summation for the bracket. The categorifications are based on new gradings
associated with these arrow numbers, and give homology theories associated with
oriented virtual knots and links via extra structure on the Khovanov chain
complex. Applications are given to the estimation of virtual crossing number
and surface genus of virtual knots and links.
  Key Words: Jones polynomial, bracket polynomial, extended bracket polynomial,
arrow polynomial, Miyazawa polynomial, Khovanov complex, Khovanov homology,
Reidemeister moves, virtual knot theory, differential, partial differential,
grading, dotted grading, vector grading.
