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J-braid groups are torus necklace groups

Igor Haladjian

TL;DR

This work extends the known link between rank-two complex reflection groups and torus knot groups to the broader class of $J$-reflection groups by introducing torus necklaces and their associated link groups. It defines $J$-braid groups via torsion-free presentations and proves a precise correspondence with the link groups of torus necklaces, with meridians mapping to braid reflections. The paper shows that torus necklace groups coincide with circular groups up to isomorphism, implying a Garside structure for these link groups and clarifying which link groups have nontrivial centers. Together, these results unify topological realizations of rank-two complex braid groups, provide explicit presentations and automorphisms, and connect to Seifert links and JSJ decompositions, yielding both algebraic and geometric insights into a broad family of groups.

Abstract

We construct a family of links we call torus necklaces for which the link groups are precisely the braid groups of generalised $J$-reflection groups. Moreover, this correspondence exhibits the meridians of the aforementioned link groups as braid reflections. In particular, this construction generalises to all irreducible rank two complex reflection groups a well-known correspondence between some rank two complex braid groups and some torus knot groups. In addition, as abstract groups, we show that the family of link groups associated to Seifert links coincides with the family of circular groups. This shows that every time a link group has a non-trivial center, it is a Garside group.

J-braid groups are torus necklace groups

TL;DR

This work extends the known link between rank-two complex reflection groups and torus knot groups to the broader class of -reflection groups by introducing torus necklaces and their associated link groups. It defines -braid groups via torsion-free presentations and proves a precise correspondence with the link groups of torus necklaces, with meridians mapping to braid reflections. The paper shows that torus necklace groups coincide with circular groups up to isomorphism, implying a Garside structure for these link groups and clarifying which link groups have nontrivial centers. Together, these results unify topological realizations of rank-two complex braid groups, provide explicit presentations and automorphisms, and connect to Seifert links and JSJ decompositions, yielding both algebraic and geometric insights into a broad family of groups.

Abstract

We construct a family of links we call torus necklaces for which the link groups are precisely the braid groups of generalised -reflection groups. Moreover, this correspondence exhibits the meridians of the aforementioned link groups as braid reflections. In particular, this construction generalises to all irreducible rank two complex reflection groups a well-known correspondence between some rank two complex braid groups and some torus knot groups. In addition, as abstract groups, we show that the family of link groups associated to Seifert links coincides with the family of circular groups. This shows that every time a link group has a non-trivial center, it is a Garside group.

Paper Structure

This paper contains 6 sections, 44 theorems, 86 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Link groups
  3. Torus necklaces
  4. Definition of torus necklaces
  5. Link groups of torus necklaces
  6. Link between torus necklace groups and circular groups

Key Result

Theorem 1.1

A group is isomorphic to a finite $J$-reflection group if and only if it is an irreducible complex reflection group of rank two.

Figures (5)

  • Figure 1: Generators of Presentation \ref{['LinkPresIntro']} as meridians
  • Figure 2: Generators of Presentation \ref{['LinkPres']} as meridians
  • Figure 3: The braids $\textbf{b}_{3,4}$, $q_2(\textbf{b}_{3,4})$, $q_5(\textbf{b}_{3,4})$ and $q(\textbf{b}_{3,4})$
  • Figure 4: The torus necklaces $\mathrm L_*^*(3,4)$, $\mathrm L_*(3,4)$, $\mathrm L^*(3,4)$ and $\mathrm L(3,4)$
  • Figure 5: The braid $(\sigma_5\sigma_4\sigma_3\sigma_2\sigma_1)(\sigma_1\sigma_2\sigma_3\sigma_4\sigma_5)$ and the key-chain link $\mathrm{K}(5)$

Theorems & Definitions (104)

  • Theorem 1.1
  • Definition 1.2: Gobet Toric
  • Theorem 1.3: ArtinZopfe, see also BirmanBook
  • Theorem 1.4: BZTorus
  • Theorem 1.5: SeifertLinks
  • Theorem 1.6: Theorem \ref{['NecklaceGroupsAreJBraidgroups']}
  • Remark 1.7: Corollary \ref{['CorollaryHomotopyEquivalence']}
  • Theorem 1.8: Theorems \ref{['NecklaceGroupsAreJBraidgroups']} and \ref{['LnIso']} and Remark \ref{['CircularIso']}
  • Corollary 1.9: Corollary \ref{['GarsideLinkGroups']}
  • Definition 2.1: AA
  • ...and 94 more