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The Mathematics of Knots pp 95–124Cite as

On Two Categorifications of the Arrow Polynomial for Virtual Knots

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Part of the book series: Contributions in Mathematical and Computational Sciences ((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.

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References

  1. Asaeda, M.M., Przytycki, J.H., Sikora, A.S.: Categorification of the Kauffman bracket skein module of I-bundles over surfaces. Algebraic Geom. Topol. 4, 1177–1210 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bar-Natan, D.: On Khovanov’s categorification of the Jones polynomial. Algebraic Geom. Topol. 2(16), 337–370 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bar-Natan, D.: Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9, 1443–1499 (2005). arXiv:math.GT/0410495

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourgoin, M.O.: Twisted link theory. Algebraic Geom. Topol. 8(3), 1249–1279 (2008). arXiv:math.GT/0608233

    Article  MATH  MathSciNet  Google Scholar 

  5. Caprau, C.: The universal sl(2) cohomology via webs and foams. arXiv:0802.2848v2 [math.GT]

  6. Champanerkar, A., Kofman, I.: Spanning trees and Khovanov homology. Proc. Am. Math. Soc. 137, 2157–2167 (2009). arXiv:math.GT/0607510

    Article  MATH  MathSciNet  Google Scholar 

  7. Clark, C., Morrison, S., Walker, K.: Fixing the functoriality of Khovanov homology. arXiv:math.GT/0701339v2 (2007)

  8. Drobotukhina, Yu.V.: An Analogue of the Jones-Kauffman polynomial for links in R 3 and a generalisation of the Kauffman-Murasugi Theorem. Algebra Anal. 2(3), 613–630 (1991)

    MATH  MathSciNet  Google Scholar 

  9. Dye, H., Kauffman, L.H.: Virtual crossing number and the arrow polynomial. J. Knot Theory Ramif. 18(10), 1335–1357 (2009). arXiv:0810.3858 [math.GT]

    Article  MATH  MathSciNet  Google Scholar 

  10. Fenn, R.A., Kauffman, L.H., Manturov, V.O.: Virtual knots: unsolved problems. In: Proceedings of the Conference “Knots in Poland-2003”. Fundamenta Mathematicae, vol. 188, pp. 293–323 (2005)

    Google Scholar 

  11. Fomenko, A.T.: The theory of multidimensional integrable hamiltonian systems (with arbitrary many degrees of freedom). Molecular table of all integrable systems with two degrees of freedom. Adv. Sov. Math. 6, 1–35 (1991)

    MathSciNet  Google Scholar 

  12. Goussarov, M., Polyak, M., Viro, O.: Finite type invariants of classical and virtual links. Topology 39, 1045–1068 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kauffman, L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kauffman, L.H.: Virtual knot theory. Eur. J. Comb. 20(7), 662–690 (1999)

    Article  MathSciNet  Google Scholar 

  15. Kauffman, L.H.: A self-linking invariant of virtual knots. Fund. Math. 184, 135–158 (2004). arXiv:math.GT/0405049

    Article  MATH  MathSciNet  Google Scholar 

  16. Kauffman, L.H.: An extended bracket polynomial for virtual knots and links. J. Knot Theory Ramif. 18(10), 1369–1422 (2009). arXiv:0712.2546 [math.GT]

    Article  MATH  MathSciNet  Google Scholar 

  17. Kauffman, L.H., Dye, H. A.: Minimal surface representations of virtual knots and links. Algebraic Geom. Topol. 5, 509–535 (2005). arXiv:math.GT/0401035

    Article  MATH  MathSciNet  Google Scholar 

  18. Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (1997)

    Article  MathSciNet  Google Scholar 

  19. Khovanov, M.: Link homology and Frobenius extensions. Fundam. Math. 190, 179–190 (2006). arXiv:math.GT/0411447

    Article  MATH  MathSciNet  Google Scholar 

  20. Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. Fundam. Math. 199(1), 1–91 (2008). arXiv:math.GT/0401268

    Article  MATH  MathSciNet  Google Scholar 

  21. Lee, E.S.: On Khovanov invariant for alternating links. arXiv:math.GT/0210213 (2003)

  22. Manturov, V.O.: Kauffman–like polynomial and curves in 2-surfaces. J. Knot Theory Ramif. 12(8), 1145–1153 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Manturov, V.O.: The Khovanov polynomial for Virtual Knots. Russ. Acad. Sci. Dokl. 398(1), 11–15 (2004)

    MathSciNet  Google Scholar 

  24. Manturov, V.O.: Minimal diagrams of classical knots. arXiv:math.GT/0501510 (2005)

  25. Manturov, V.O.: Teoriya Uzlov. RCD, M.-Izhevsk (2005). (Knot Theory, In Russian)

    Google Scholar 

  26. Manturov, V.O.: The Khovanov complex for virtual links. Fundam. Appl. Math. 11(4), 127–152 (2005)

    Google Scholar 

  27. Manturov, V.O.: The Khovanov complex and minimal knot diagrams. Russ. Acad. Sci. Dokl. 406(3), 308–311 (2006)

    MathSciNet  Google Scholar 

  28. Manturov, V.O.: Additional gradings in the Khovanov complex for thickened surfaces. Dokl. Math. 77, 368–370 (2007)

    Article  Google Scholar 

  29. Manturov, V.O.: Khovanov homology for virtual knots with arbitrary coefficients. Russ. Acad. Sci. Izv. 71(5), 111–148 (2007)

    MathSciNet  Google Scholar 

  30. Manturov, V.O.: Additional gradings in Khovanov homology. In: Proceedings of the Conference “Topology and Physics. Dedicated to the Memory of Xiao-Song Lin”. Nankai Tracts in Mathematics, pp. 266–300. World Scientific, Singapore (2008)

    Google Scholar 

  31. Manturov, V.O.: On free knots and links. arXiv:0902.0127 [math.GT] (2009)

  32. Manturov, V.O.: On free Knots. arXiv:0901.2214 [math.GT] (2009)

  33. Miyazawa, Y.: Magnetic Graphs and an Invariant for Virtual Links. J. Knot Theory Ramif. 15(10), 1319–1334 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Miyazawa, Y.: A multi-variable polynomial invariant for virtual knots and links. J. Knot Theory Ramif. 17(11), 1311–1326 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ozsváth, P., Rasmussen, J., Szabó, Z.: Odd Khovanov homology. arXiv:0710.4300 [math-qa] (2007)

  36. Rasmussen, J.A.: Khovanov homology and the slice genus. arXiv:math.GT/0402131 (2004)

  37. Turaev, V.G.: A simple proof of the Murasugi and Kauffman theorems on alternating links. Enseign. Math. (2), 33(3–4), 203–225 (1987)

    MATH  MathSciNet  Google Scholar 

  38. Viro, O.: Khovanov homology, its definitions and ramifications. Fundam. Math. 184, 317–342 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  39. Viro, O.: Remarks on definition of Khovanov homology. math.GT/0202199

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Author information

Authors and Affiliations

  1. Division of Science and Mathematics, McKendree University, 701 College Road, Lebanon, IL, 62254, USA

    Heather Ann Dye

  2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA

    Louis Hirsch Kauffman

  3. People’s Friendship University of Russia, Miklukho-Maklay Street, 6, Moscow, 117198, Russia

    Vassily Olegovich Manturov

Authors
  1. Heather Ann Dye
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  2. Louis Hirsch Kauffman
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  3. Vassily Olegovich Manturov
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Corresponding author

Correspondence to Heather Ann Dye .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Heidelberg, Im Neuenheimer Feld 288, Heidelberg, 69120, Germany

    Markus Banagl

  2. Department of Mathematics, University of Heidelberg, Im Neuenheimer Feld 288, Heidelberg, 69120, Germany

    Denis Vogel

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© 2011 Springer-Verlag Berlin Heidelberg

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Dye, H.A., Kauffman, L.H., Manturov, V.O. (2011). On Two Categorifications of the Arrow Polynomial for Virtual Knots. In: Banagl, M., Vogel, D. (eds) The Mathematics of Knots. Contributions in Mathematical and Computational Sciences, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15637-3_4

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