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On some torus knot groups and submonoids of the braid groups. (English) Zbl 07552695

J. Algebra 607, Part B, 260-289 (2022); addendum ibid. 633, 666-667 (2023).
Summary: The submonoid of the 3-strand braid group \(\mathcal{B}_3\) generated by \(\sigma_1\) and \(\sigma_1 \sigma_2\) is known to yield an exotic Garside structure on \(\mathcal{B}_3\). We introduce and study an infinite family \(( M_n )_{n \geq 1}\) of Garside monoids generalizing this exotic Garside structure, i.e., such that \(M_2\) is isomorphic to the above monoid. The corresponding Garside group \(G( M_n)\) is isomorphic to the \((n, n + 1)\)-torus knot group-which is isomorphic to \(\mathcal{B}_3\) for \(n = 2\) and to the braid group of the exceptional complex reflection group \(G_{12}\) for \(n = 3\). This yields a new Garside structure on \((n, n + 1)\)-torus knot groups, which already admit several distinct Garside structures. The \((n, n + 1)\)-torus knot group is an extension of \(\mathcal{B}_{n + 1} \), and the Garside monoid \(M_n\) surjects onto the submonoid \({\Sigma}_n\) of \(\mathcal{B}_{n + 1}\) generated by \(\sigma_1, \sigma_1 \sigma_2, \ldots, \sigma_1 \sigma_2 \cdots \sigma_n\), which is not a Garside monoid when \(n > 2\). Using a new presentation of \(\mathcal{B}_{n + 1}\) that is similar to the presentation of \(G( M_n)\), we nevertheless check that \({\Sigma}_n\) is an Ore monoid with group of fractions isomorphic to \(\mathcal{B}_{n + 1} \), and give a conjectural presentation of it, similar to the defining presentation of \(M_n\). This partially answers a question of Dehornoy-Digne-Godelle-Krammer-Michel.

MSC:

20-XX Group theory and generalizations
16-XX Associative rings and algebras
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[1] Artin, E., Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamb., 4, 1, 47-72 (1925) · JFM 51.0450.01
[2] Bessis, D., The dual braid monoid, Ann. Sci. Éc. Norm. Supér., 36, 647-683 (2003) · Zbl 1064.20039
[3] Bessis, D., A dual braid monoid for the free group, J. Algebra, 302, 275-309 (2006)
[4] Bessis, D., Finite complex reflection arrangements are \(K(\pi, 1)\), Ann. Math., 181, 1, 55-69 (2015)
[5] Bessis, D.; Digne, F.; Michel, J., Springer theory in braid groups and the Birman-Ko-Lee monoid, Pac. J. Math., 205, 287-309 (2002) · Zbl 1056.20023
[6] Birman, J.; Brendle, T., Braids: a survey, (Handbook of Knot Theory (2005), Elsevier B.V.: Elsevier B.V. Amsterdam), 19-103 · Zbl 1094.57006
[7] Birman, J.; Ko, K. H.; Lee, S. J., A new approach to the word and conjugacy problems in the braid groups, Adv. Math., 139, 322-353 (1998) · Zbl 0937.20016
[8] Brav, C.; Thomas, H., Braid groups and Kleinian singularities, Math. Ann., 351, 4, 1005-1017 (2011) · Zbl 1264.14026
[9] Brieskorn, E.; Saito, K., Artin-Gruppen und Coxeter-Gruppen, Invent. Math., 17, 245-271 (1972) · Zbl 0243.20037
[10] Broué, M.; Malle, G.; Rouquier, R., Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math., 500, 127-190 (1998) · Zbl 0921.20046
[11] Clifford, A. H.; Preston, G. B., The Algebraic Theory of Semigroups, Vol. II, Mathematical Surveys, vol. 7 (1967), American Mathematical Society: American Mathematical Society Providence, R.I. · Zbl 0178.01203
[12] Corran, R.; Lee, E.-K.; Lee, S.-J., Braid groups of imprimitive complex reflection groups, J. Algebra, 427, 387-425 (2015) · Zbl 1320.20037
[13] Coxeter, H. S.M., Factor groups of the braid group, (Proc. 4th Canad. Math. Cong. (1959)), 95-122 · Zbl 0093.25003
[14] Dehornoy, P., Groupes de Garside, Ann. Sci. Éc. Norm. Supér. (4), 35, 2, 267-306 (2002) · Zbl 1017.20031
[15] Dehornoy, P., The subword reversing method, Int. J. Algebra Comput., 21, 1-2, 71-118 (2011) · Zbl 1256.20053
[16] Dehornoy, P., A cancellativity criterion for presented monoids, Semigroup Forum, 99, 2, 368-390 (2019) · Zbl 1467.20047
[17] Dehornoy, P.; Digne, F.; Krammer, D.; Godelle, E.; Michel, J., Foundations of Garside Theory, Tracts in Mathematics, vol. 22 (2015), Europ. Math. Soc. · Zbl 1370.20001
[18] Dehornoy, P.; Paris, L., Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. Lond. Math. Soc. (3), 79, 3, 569-604 (1999) · Zbl 1030.20021
[19] Deligne, P., Les immeubles des groupes de tresses généralisés, Invent. Math., 17, 273-302 (1972) · Zbl 0238.20034
[20] Digne, F., Présentations duales des groupes de tresses de type affine \(\widetilde{A} \), Comment. Math. Helv., 81, 1, 23-47 (2006) · Zbl 1143.20020
[21] Digne, F., A Garside presentation for Artin groups of type \(\widetilde{C}_n\), Ann. Inst. Fourier, 62, 2, 641-666 (2012) · Zbl 1260.20056
[22] Garside, F. A., The braid group and other groups, Quart. J. Math. Oxford Ser., 20, 2, 235-254 (1969) · Zbl 0194.03303
[23] Han, J. W.; Ko, K. H., Positive presentations of the braid groups and the embedding problem, Math. Z., 240, 1, 211-232 (2002) · Zbl 1078.20039
[24] Jensen, L. T., The 2-braid group and Garside normal form, Math. Z., 286, 1-2, 491-520 (2017) · Zbl 1423.20031
[25] Kassel, C.; Turaev, V., Braid Groups, Graduate Texts in Mathematics, vol. 247 (2008), Springer: Springer New York · Zbl 1208.20041
[26] Krammer, D., Braid groups are linear, Ann. Math. (2), 155, 1, 131-156 (2002) · Zbl 1020.20025
[27] Licata, T.; Queffelec, H., Braid groups of type ADE, Garside structures, and the categorified root lattice (2017), preprint
[28] Picantin, M., Petits groupes gaussiens (2000), Université de Caen, PhD Thesis
[29] Picantin, M., Automatic structures for torus link groups, J. Knot Theory Ramif., 12, 6, 833-866 (2003) · Zbl 1063.20047
[30] Rolfsen, D., Knots and Links, Mathematics Lecture Series, vol. 7 (1976), Publish or Perish, Inc.: Publish or Perish, Inc. Berkeley, Calif., 439 pp · Zbl 0339.55004
[31] Sergiescu, V., Graphes planaires et présentations des groupes de tresses, Math. Z., 214, 3, 477-490 (1993) · Zbl 0819.20040
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