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On Two Categorifications of the Arrow Polynomial for Virtual Knots

  • Heather Ann DyeEmail author
  • Louis Hirsch Kauffman
  • Vassily Olegovich Manturov
Chapter
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 1)

Abstract

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 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.

Keywords

Partial Differential Virtual Link Jones Polynomial Circle Graph Reidemeister Move 
These 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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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Heather Ann Dye
    • 1
    Email author
  • Louis Hirsch Kauffman
    • 2
  • Vassily Olegovich Manturov
    • 3
  1. 1.Division of Science and MathematicsMcKendree UniversityLebanonUSA
  2. 2.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.People’s Friendship University of RussiaMoscowRussia

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