On some torus knot groups and submonoids of the braid groups
| Autor: | Thomas Gobet |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Monoid
Algebra and Number Theory Complex reflection group Group (mathematics) 010102 general mathematics Braid group Structure (category theory) Mathematics::Geometric Topology 01 natural sciences Torus knot Combinatorics Mathematics::Group Theory 0103 physical sciences 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | Journal of Algebra. 607:260-289 |
| ISSN: | 0021-8693 |
| DOI: | 10.1016/j.jalgebra.2021.04.014 |
| Popis: | The submonoid of the 3-strand braid group B 3 generated by σ 1 and σ 1 σ 2 is known to yield an exotic Garside structure on B 3 . We introduce and study an infinite family ( M n ) n ≥ 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 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 B n + 1 , and the Garside monoid M n surjects onto the submonoid Σ n of B n + 1 generated by σ 1 , σ 1 σ 2 , … , σ 1 σ 2 ⋯ σ n , which is not a Garside monoid when n > 2 . Using a new presentation of B n + 1 that is similar to the presentation of G ( M n ) , we nevertheless check that Σ n is an Ore monoid with group of fractions isomorphic to 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. |
| Databáze: | OpenAIRE |
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