Advertisement

Arc Shift Number for Some Virtual Knots

  • K. Kaur
  • A. Gill
  • M. PrabhakarEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this paper, we compute the arc shift number for some classes of virtual knots and show that for every positive integer n, there exist infinitely many virtual knots with arc shift number n. We conclude the paper by computing the arc shift number for an infinite family of virtual knots with virtual bridge index one.

Keywords

Virtual knot Odd writhe Arc shift Virtual bridge number 

2010 Mathematics Subject Classifications

57M25 57M27 

References

  1. 1.
    E. Byberi, V. Chernov, Virtual bridge number one knots. Commun. Contemp. Math. 10(suppl. 1), 1013–1021 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Z. Cheng, H. Gao, A polynomial invariant of virtual links. J. Knot Theory Ramif. 22(12), 1341002 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Conway, An enumeration of knots and links, and some of their algebraic properties. in Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), p. 329–358 (1970)Google Scholar
  4. 4.
    L. Folwaczny, L. Kauffman, A linking number definition of the affine index polynomial and applications. J. Knot Theory Ramif. 22(12), 1341004 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Kauffman, An affine index polynomial invariant of virtual knots. J. Knot Theory Ramif. 22(04), 1340007, 30pp (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Kauffman, Virtual knot theory. Eur. J. Combin. 20(7), 663–690 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Kauffman, A self-linking invariant of virtual knots. Fundam. Math. 184, 135–158 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    K. Kaur, A. Gill, M. Prabhakar, Arc shift number and region arc shift number for virtual knots (2018), arXiv:1808.04172
  9. 9.
    K. Kaur, M. Prabhakar, A. Vesnin, Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants. J. Knot Theory Ramif. 27(13), 1842015 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Satoh, K. Taniguchi, The writhes of a virtual knot. Fundam. Math. 225, 327–341 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoparRupnagarIndia

Personalised recommendations