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A lexicographic section of the braid arrangement and the modified Artin presentation

So Yamagata

TL;DR

This work connects geometric braid monodromy to the modified Artin presentation by constructing a lexicographic 2–section of the braid arrangement and performing explicit braid monodromy computations to obtain a presentation of the pure braid group that matches Margalit–McCammond’s modified Artin form. It then extends the approach to Manin–Schechtman arrangements MS(n,k), and concretely treats the case k=2 to derive an explicit π_1(\\mathbb{C}^n \\setminus MS(n,2)) presentation with higher–multiplicity generators and relations, thereby generalizing the modified Artin structure to higher Bruhat–type configurations. The results provide a geometrically natural explanation for the modified Artin presentation and demonstrate how braid monodromy can systematically produce explicit, combinatorially driven presentations in higher dimensions. Overall, the paper builds a constructive bridge between hyperplane arrangement topology, braid monodromy, and higher Bruhat-type groups, with potential implications for understanding non-uniqueness phenomena in higher braid groups.

Abstract

We study a specific line arrangement obtained from a generic $2$-section of the braid arrangement, and compute the fundamental group of its complement via braid monodromy. We show that the resulting presentation of the fundamental group coincides, under the identification of generators, with the modified Artin presentation introduced by Margalit and McCammond. Moreover, we extend the construction to the Manin--Schechtman arrangements $MS(n, k)$, which are higher analogues of the braid arrangement. Focusing on the case $k = 2$, we obtain an explicit presentation of $π_1(\mathbb{C}^n \setminus MS(n, 2))$.

A lexicographic section of the braid arrangement and the modified Artin presentation

TL;DR

This work connects geometric braid monodromy to the modified Artin presentation by constructing a lexicographic 2–section of the braid arrangement and performing explicit braid monodromy computations to obtain a presentation of the pure braid group that matches Margalit–McCammond’s modified Artin form. It then extends the approach to Manin–Schechtman arrangements MS(n,k), and concretely treats the case k=2 to derive an explicit π_1(\\mathbb{C}^n \\setminus MS(n,2)) presentation with higher–multiplicity generators and relations, thereby generalizing the modified Artin structure to higher Bruhat–type configurations. The results provide a geometrically natural explanation for the modified Artin presentation and demonstrate how braid monodromy can systematically produce explicit, combinatorially driven presentations in higher dimensions. Overall, the paper builds a constructive bridge between hyperplane arrangement topology, braid monodromy, and higher Bruhat-type groups, with potential implications for understanding non-uniqueness phenomena in higher braid groups.

Abstract

We study a specific line arrangement obtained from a generic -section of the braid arrangement, and compute the fundamental group of its complement via braid monodromy. We show that the resulting presentation of the fundamental group coincides, under the identification of generators, with the modified Artin presentation introduced by Margalit and McCammond. Moreover, we extend the construction to the Manin--Schechtman arrangements , which are higher analogues of the braid arrangement. Focusing on the case , we obtain an explicit presentation of .
Paper Structure (9 sections, 6 theorems, 39 equations, 10 figures)

This paper contains 9 sections, 6 theorems, 39 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Modified Artin presentation for the pure braid group
  3. Braid monodromy technique
  4. Line arrangement and its wiring diagram
  5. Computation of the fundamental group
  6. Braid arrangement and braid monodromy
  7. A lexicographic line arrangement
  8. Computation of braid monodromy and presentation for the pure braid group
  9. Manin--Schechtman arrangements

Key Result

Theorem 2.6

Let $D_K$ be a convexly punctured disc. Then $PB_K$ is generated by the convex swings $S_{ij}$, and all defining relations among them fall into the following three types (see Figure fig:puc_conf):

Figures (10)

  • Figure 1: The configurations of punctures corresponding to (R-I)--(R-III), from left to right.
  • Figure 2: The line arrangement of six lines with seven intersection points.
  • Figure 3: The wiring diagram of the line arrangement in Figure \ref{['fig:exlin']}.
  • Figure 4: Action of Lefschetz pairs on $s_3^{\mathrm{init}}$.
  • Figure 5: Skeletons for the non--crossing, crossing, and admissible types, from left to right.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Theorem 2.3 in MM09
  • Definition 4.1
  • Definition 4.2
  • Proposition 4.3
  • Theorem 4.4
  • ...and 7 more