Abstract
AbstractTwo 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 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.KeywordsPartial DifferentialVirtual LinkJones PolynomialCircle GraphReidemeister MoveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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