Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.

Keywords:

AMSC: 57K12

References

1. C. Adams , Triple crossing number of knots and links, J. Knot Theory Ramifications 22 (2013) 1350006. Link, Web of Science, Google Scholar
2. C. Adams , Quadruple crossing number of knots and links, Math. Proc. Cambridge Philos. Soc. 156(2) (2014) 241–253. Web of Science, Google Scholar
3. C. Adams, O. Capovilla-Searle, J. Freeman, D. Irvine, S. Petti, D. Vitek, A. Weber and S. Zhang , Bounds on ubercrossing and petal number for knots, J. Knot Theory Ramifications 24(2) (2015) 1550012. Link, Web of Science, Google Scholar
4. C. Adams, O. Capovilla-Searle, J. Freeman, D. Irvine, S. Petti, D. Vitek, A. Weber and S. Zhang , Multicrossing numbers and the span of the bracket polynomial, Math. Proc. Cambridge Philos. Soc. 164(1) (2018) 147–178. Web of Science, Google Scholar
5. C. Adams, T. Crawford, B. DeMeo, M. Landry, M. Montee, S. Park, S. Venkatesh and F. Yhee , Knot projections with a single multi-crossing, J. Knot Theory Ramifications 24(3) (2015) 1550011. Link, Web of Science, Google Scholar
6. C. Adams, J. Hoste and M. Palmer , Triple crossing number and moves on triple-crossing link diagrams, J. Knot Theory Ramifications 28(11) (2019) 1940001. Link, Web of Science, Google Scholar
7. L. Colton, C. Glover, M. Hughes and S. Sandberg , A Reidemeister type theorem for petal diagrams of knots, Topol. Appl. 267 (2019) 106896. Web of Science, Google Scholar
8. C. Even-Zohar, J. Hass, N. Linial and T. Nowik , The distribution of knots in the petaluma model, Algebr. Geom. Topol. 18(6) (2018) 3647–3667. Web of Science, Google Scholar
9. C. Even-Zohar, J. Hass, N. Linial and T. Nowik , Invariants of random knots and links, Discrete Comput. Geom. 56(2) (2016) 274–314. Web of Science, Google Scholar
10. P. Flajolet and R. Sedgewick , Analytic Combinatorics (Cambridge University Press, 2009), http://algo.inria.fr/flajolet/Publications/AnaCombi. Google Scholar
11. OEIS Foundation Inc., The on-line encyclopedia of integer sequences (2019), http://oeis.org. Google Scholar
12. V. Jones, Planar algebras I, preprint (1999), arXiv:9909027. Google Scholar
13. T. Kanenobu , Forbidden moves unknot a virtual knot, J. Knot Theory Ramifications 10(1) (2001) 89–96. Link, Web of Science, Google Scholar
14. L. Kauffman , Virtual knot theory, Eur. J. Comb. 20 (1999) 663–691. Web of Science, Google Scholar
15. V. O. Manturov , Parity and projection from virtual knots to classical knots, J. Knot Theory Ramifications 22 (2013) 1350044. Link, Web of Science, Google Scholar
16. S. Nelson , Unknotting virtual knots with Gauss diagram forbidden moves, J. Knot Theory Ramifications 10(6) (2001) 931–935. Link, Web of Science, Google Scholar
17. S. Yoshiike and K. Ichihara , Forbidden detour number on virtual knot, Kyungpook Math. J. 61 (2021) 205–212. Web of Science, Google Scholar