The Alternating Normal Form of Braids and Its Minimal Automaton

Authors Vincent Jugé , June Roupin



PDF
Thumbnail PDF

File

LIPIcs.AofA.2024.23.pdf
  • Filesize: 0.76 MB
  • 15 pages

Document Identifiers

Author Details

Vincent Jugé
  • LIGM, CNRS & Univ Gustave Eiffel, Marne-la-Vallée, France
  • IRIF, CNRS & Université Paris Cité, France
June Roupin
  • LIGM, CNRS & Univ Gustave Eiffel, Marne-la-Vallée, France

Cite AsGet BibTex

Vincent Jugé and June Roupin. The Alternating Normal Form of Braids and Its Minimal Automaton. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.23

Abstract

The alternating normal form of braids is a well-known normal form on standard braid monoids. This normal form is regular: the language it identifies with is regular. We give a characterisation of the minimal automaton of this language and compute its size, both in terms of number of states and of transitions, depending on the number of generators of the monoid.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
Keywords
  • Automata
  • braids
  • enumeration
  • normal forms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Sergei Adian. Fragments of the word Δ in the braid group. Matematicheskie Zametki, 36(1):25-34, 1984. URL: https://doi.org/10.1007/BF01139549.
  2. Emil Artin. Theorie der Zöpfe. In Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, volume 4, pages 47-72. Springer, 1925. URL: https://doi.org/10.1007/BF02950718.
  3. Serge Burckel. The wellordering on positive braids. Journal of Pure and Applied Algebra, 120(1):1-17, 1997. URL: https://doi.org/10.1016/S0022-4049(96)00072-2.
  4. Ruth Charney. Artin groups of finite type are biautomatic. Mathematische Annalen, 292(1):671-683, 1992. URL: https://doi.org/10.1007/BF01444642.
  5. Patrick Dehornoy. Alternating normal forms for braids and locally Garside monoids. Journal of pure and applied algebra, 212(11):2413-2439, 2008. URL: https://doi.org/10.1016/j.jpaa.2008.03.027.
  6. Patrick Dehornoy, François Digne, Eddy Godelle, Daan Krammer, and Jean Michel. Foundations of Garside theory, volume 22. Citeseer, 2015. URL: https://doi.org/10.4171/139.
  7. Pierre Deligne. Les immeubles des groupes de tresses généralisés. Inventiones mathematicae, 17:273-302, 1972. URL: https://doi.org/10.1007/BF01406236.
  8. David Epstein. Word processing in groups. CRC Press, 1992. URL: https://doi.org/10.1201/9781439865699.
  9. Ramón Flores and Juan González-Meneses. On lexicographic representatives in braid monoids. Journal of Algebraic Combinatorics, 52(4):561-597, 2020. URL: https://doi.org/10.1007/s10801-019-00913-7.
  10. Ramón Flores and Juan González-Meneses. On the growth of Artin-Tits monoids and the partial theta function. Journal of Combinatorial Theory, Series A, 190:105623, 2022. URL: https://doi.org/10.1016/j.jcta.2022.105623.
  11. Jean Fromentin. The rotating normal form of braids is regular. Journal of Algebra, 501:545-570, 2018. URL: https://doi.org/10.1016/j.jalgebra.2018.01.001.
  12. Frank Garside. The braid group and other groups. The Quarterly Journal of Mathematics, 20(1):235-254, 1969. URL: https://doi.org/10.1093/qmath/20.1.235.
  13. Volker Gebhardt and Juan González-Meneses. Generating random braids. Journal of Combinatorial Theory, Series A, 120(1):111-128, 2013. URL: https://doi.org/10.1016/j.jcta.2012.07.003.
  14. Jean Michel. A note on words in braid monoids. Journal of Algebra, 215(1):366-377, 1999. URL: https://doi.org/10.1006/jabr.1998.7723.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail