Lower Bounds on Virtual Crossing Number and Minimal Surface Genus

  • Kumud Bhandari
  • H. A. DyeEmail author
  • Louis H. Kauffman
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 1)


We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.


Minimal Genus Virtual Link Dehn Twist Reidemeister Move Link Diagram 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AM]
    Afanasiev, D., Manturov, V.O.: On virtual crossing number estimates for virtual links. J. Knot Theory Ramif. 18(6), 757–772 (2009). arXiv:0811.0712 zbMATHCrossRefMathSciNetGoogle Scholar
  2. [CKS02]
    Carter, J.S., Kamada, S., Saito, M.: Stable equivalence of knots on surfaces and virtual knot cobordisms. (English summary) Knots 2000 Korea, vol. 1 (Yongpyong). J. Knot Theory Ramif. 11(3), 311–322 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  3. [DK]
    Dye, H.A., Kauffman, L.H.: Virtual crossing number and the arrow polynomial. J. Knot Theory Ramif. 18(10), 1335–1357 (2009). arXiv:0810.3858 zbMATHCrossRefMathSciNetGoogle Scholar
  4. [DK05]
    Dye, H.A., Kauffman, L.H.: Minimal surface representations of virtual knots and links. Algebraic Geom. Topol. 5, 509–535 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. [FKM05]
    Fenn, R., Kauffman, L.H., Manturov, V.O.: Virtual knot theory—unsolved problems. Fund. Math. 188, 293–323 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Gre]
    Green, J.: A table of virtual knots.
  7. [Kam07a]
    Kamada, N.: An index of an enhanced state of a virtual link diagram and Miyazawa polynomials. Hiroshima Math. J. 37(3), 409–429 (2007) zbMATHMathSciNetGoogle Scholar
  8. [Kam07b]
    Kamada, N.: Some relations on Miyazawa’s virtual knot invariant. Topol. Appl. (7), 1417–1429 (2007) Google Scholar
  9. [Kau]
    Kauffman, L.H.: An extended bracket polynomial for virtual knots and links. J. Knot Theory Ramif. 18(10), 1369–1422 (2009). arXiv:0712.2546 zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Kau99]
    Kauffman, L.H.: Virtual knot theory. Eur. J. Combinatorics (7), 663–690 (1999) Google Scholar
  11. [Kau01]
    Kauffman, L.H.: Detecting virtual knots. Atti. Sem. Mat. Fis. Univ. Modena Suppl. IL, 241–282 (2001) MathSciNetGoogle Scholar
  12. [KM05]
    Kamada, N., Miyazawa, Y.: A 2-variable polynomial invariant for a virtual link derived from magnetic graphs. Hiroshima Math. J. 35(2), 309–326 (2005) zbMATHMathSciNetGoogle Scholar
  13. [Kup03]
    Kuperberg, G.: What is a virtual link? Algebraic. Geom. Topol. 3, 587–591 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  14. [Man03]
    Manturov, V.O.: Kauffman–like polynomial and curves in 2-surfaces. J. Knot Theory Ramif. 8, 1145–1153 (2003) CrossRefMathSciNetGoogle Scholar
  15. [Miy08]
    Miyazawa, Y.: A multi-variable polynomial invariant for virtual knots and links. J. Knot Theory Ramif. 17(11), 1311–1326 (2008) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kumud Bhandari
    • 1
  • H. A. Dye
    • 1
    Email author
  • Louis H. Kauffman
    • 2
  1. 1.Division of Science and MathematicsMcKendree UniversityLebanonUSA
  2. 2.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

Personalised recommendations