Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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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.

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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

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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)

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

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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)

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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.

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FIGURE 15

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

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

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FIGURE 18. R3 move realized via arc shift moves

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FIGURE 19

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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

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FIGURE 22. Unknotting virtual trefoil using arc shift move

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FIGURE 23

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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

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FIGURE 28. Forbidden detour move

DBSHBB_2019_v56n4_1063_f0029.png 이미지

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

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