No Access

Virtual tribrackets

Sam Nelson

Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Ave., Claremont, 91711 CA, USA

E-mail Address: Sam.Nelson@cmc.edu

Corresponding author.

Search for more papers by this author

and

Shane Pico

Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Ave., Claremont, 91711 CA, USA

E-mail Address: SPico18@students.claremontmckenna.edu

Search for more papers by this author

https://doi.org/10.1142/S0218216519500263Cited by:3 (Source: Crossref)

Abstract

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.

Keywords:

AMSC: 57M27, 57M25

References

1. D. Bar-Natan, The knot atlas, http://katlas.org/wiki/Main_Page. Google Scholar
2. J. S. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11(3) (2002) 311–322; Knots 2000 Korea, Vol. 1 (Yongpyong, 2000). Google Scholar
3. J. Ceniceros and S. Nelson, Virtual Yang-Baxter cocycle invariants, Trans. Amer. Math. Soc. 361(10) (2009) 5263–5283. Crossref, Web of Science, Google Scholar
4. W. Choi, D. Needell and S. Nelson, Boltzmann enhancements of biquasile counting invariants, J. Knot Theory Ramifications 27(14) (2018) 1850068, 12 pp. Link, Web of Science, Google Scholar
5. N. Kamada and S. Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9(1) (2000) 93–106. Link, Web of Science, Google Scholar
6. L. H. Kauffman, Virtual knot theory, European J. Combin. 20(7) (1999) 663–690. Crossref, Web of Science, Google Scholar
7. L. H. Kauffman and V. O. Manturov, Virtual biquandles, Fund. Math. 188 (2005) 103–146. Crossref, Web of Science, Google Scholar
8. J. Kim and S. Nelson, Biquasile colorings of oriented surface-links, Topol. Appl. 236 (2018) 64–76. Crossref, Web of Science, Google Scholar
9. G. Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003) 587–591. Crossref, Web of Science, Google Scholar
10. D. Needell and S. Nelson, Biquasiles and dual graph diagrams, J. Knot Theory Ramifications 26(8) (2017) 1750048. Link, Web of Science, Google Scholar
11. M. Niebrzydowski, On some ternary operations in knot theory, Fund. Math. 225(1) (2014) 259–276. Crossref, Web of Science, Google Scholar
12. M. Niebrzydowski, Homology of ternary algebras yielding invariants of knots and knotted surfaces, arXiv:1706.04307. Google Scholar
13. J. Sawollek, On Alexander–Conway polynomials for virtual knots and links, arXiv:math/9912173. Google Scholar