DOI QR코드

DOI QR Code

ARC SHIFT NUMBER AND REGION ARC SHIFT NUMBER FOR VIRTUAL KNOTS

  • Gill, Amrendra (Department of Mathematics Indian Institute of Technology Ropar) ;
  • Kaur, Kirandeep (Department of Mathematics Indian Institute of Technology Ropar) ;
  • Madeti, Prabhakar (Department of Mathematics Indian Institute of Technology Ropar)
  • Received : 2018.09.09
  • Accepted : 2019.02.07
  • Published : 2019.07.01

Abstract

In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtual knot diagram can be turned into trivial knot using arc shift (region arc shift) moves. Based upon the arc shift move and region arc shift move, we define two virtual knot invariants, arc shift number and region arc shift number respectively.

DBSHBB_2019_v56n4_1063_f0001.png 이미지

FIGURE 1. Classical and Virtual crossings

DBSHBB_2019_v56n4_1063_f0002.png 이미지

FIGURE 2. Generalized Reidemeister moves

DBSHBB_2019_v56n4_1063_f0003.png 이미지

FIGURE 3. Detour move

DBSHBB_2019_v56n4_1063_f0004.png 이미지

FIGURE 4. Local writhe or sign of a crossing

DBSHBB_2019_v56n4_1063_f0005.png 이미지

FIGURE 5. Gauss diagram corresponding to virtual figure eight knot

DBSHBB_2019_v56n4_1063_f0006.png 이미지

FIGURE 6. Reidemeister moves on Gauss diagram

DBSHBB_2019_v56n4_1063_f0007.png 이미지

FIGURE 7. Forbidden moves Fu and Fo

DBSHBB_2019_v56n4_1063_f0008.png 이미지

FIGURE 8. Arc (a,b) and (e,f)

DBSHBB_2019_v56n4_1063_f0009.png 이미지

FIGURE 9. Equivalent Arcs (a,b) and (c,d)

DBSHBB_2019_v56n4_1063_f0010.png 이미지

FIGURE 10. Arc shift on arc (a,b)

DBSHBB_2019_v56n4_1063_f0011.png 이미지

FIGURE 11. Equivalent diagrams corresponding to arc shift on arc (a,b)

DBSHBB_2019_v56n4_1063_f0012.png 이미지

FIGURE 12. Arc shift reverses orientation of arc (a,b) in D(a,b)

DBSHBB_2019_v56n4_1063_f0013.png 이미지

FIGURE 13. Arc shift move on arc (a,b)

DBSHBB_2019_v56n4_1063_f0014.png 이미지

FIGURE 14. Gauss diagram analogues to the arc shift moves shown in Fig. 13.

DBSHBB_2019_v56n4_1063_f0015.png 이미지

FIGURE 15

DBSHBB_2019_v56n4_1063_f0016.png 이미지

FIGURE 16. Arc shift move twice on same arc results in equivalent diagram

DBSHBB_2019_v56n4_1063_f0017.png 이미지

FIGURE 17. sign(c') = - sign(c)

DBSHBB_2019_v56n4_1063_f0018.png 이미지

FIGURE 18. R3 move realized via arc shift moves

DBSHBB_2019_v56n4_1063_f0019.png 이미지

FIGURE 19

DBSHBB_2019_v56n4_1063_f0020.png 이미지

FIGURE 20. ∆-move realized using arc shift moves

DBSHBB_2019_v56n4_1063_f0021.png 이미지

FIGURE 21. Turning a Gauss diagram into parallel chord diagram

DBSHBB_2019_v56n4_1063_f0022.png 이미지

FIGURE 22. Unknotting virtual trefoil using arc shift move

DBSHBB_2019_v56n4_1063_f0023.png 이미지

FIGURE 23

DBSHBB_2019_v56n4_1063_f0024.png 이미지

FIGURE 24. Region arc shift on region R1 and R2

DBSHBB_2019_v56n4_1063_f0025.png 이미지

FIGURE 25. Realizing forbidden move Fu using region arc shift

DBSHBB_2019_v56n4_1063_f0026.png 이미지

FIGURE 26. Region arc shift number is 1 for both knots

DBSHBB_2019_v56n4_1063_f0027.png 이미지

FIGURE 27

DBSHBB_2019_v56n4_1063_f0028.png 이미지

FIGURE 28. Forbidden detour move

DBSHBB_2019_v56n4_1063_f0029.png 이미지

FIGURE 29. FD move via region arc shift at region R

References

  1. A. S. Crans, B. Mellor, and S. Ganzell, The forbidden number of a knot, Kyungpook Math. J. 55 (2015), no. 2, 485-506. https://doi.org/10.5666/KMJ.2015.55.2.485
  2. M. Goussarov, M. Polyak, and O. Viro, Finite-type invariants of classical and virtual knots, Topology 39 (2000), no. 5, 1045-1068. https://doi.org/10.1016/S0040-9383(99)00054-3
  3. T. Kanenobu, Forbidden moves unknot a virtual knot, J. Knot Theory Ramifications 10 (2001), no. 1, 89-96. https://doi.org/10.1142/S0218216501000731
  4. L. H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663-690. https://doi.org/10.1006/eujc.1999.0314
  5. L. H. Kauffman, A self-linking invariant of virtual knots, Fund. Math. 184 (2004), 135-158. https://doi.org/10.4064/fm184-0-10
  6. H. Murakami and Y. Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989), no. 1, 75-89. https://doi.org/10.1007/BF01443506
  7. S. Nelson, Unknotting virtual knots with Gauss diagram forbidden moves, J. Knot Theory Ramifications 10 (2001), no. 6, 931-935. https://doi.org/10.1142/S0218216501001244
  8. M. Sakurai, An affne index polynomial invariant and the forbidden move of virtual knots, J. Knot Theory Ramifications 25 (2016), no. 7, 1650040, 13 pp.
  9. A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Japan 66 (2014), no. 3, 693-708. https://doi.org/10.2969/jmsj/06630693