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Positive Artin Presentations

Lorena Armas-Sanabria, Mario Eudave-Muñoz, Juan Pablo Díaz-González, Gabriela Hinojosa-Palafox

Abstract

An integral framed, closed pure n-braid B' in the 3-sphere describes a positive Artin presentation, if the braid B can be put on a disk with holes such that each relation describes a positive path and these paths are disjoint. In the present paper we classify the closed, pure n-braids B' in the 3-sphere, such that B represents a positive Artin presentation. Also we prove that if such B describes a positive Artin presentation then B' is strongly invertible. Such positive Artin presentation give us the fundamental group of closed, connected and orientable 3-manifolds M, and in fact, by giving an example, we show that there exist 3-manifolds whose fundamental group does not admit a positive Artin presentation.

Positive Artin Presentations

Abstract

An integral framed, closed pure n-braid B' in the 3-sphere describes a positive Artin presentation, if the braid B can be put on a disk with holes such that each relation describes a positive path and these paths are disjoint. In the present paper we classify the closed, pure n-braids B' in the 3-sphere, such that B represents a positive Artin presentation. Also we prove that if such B describes a positive Artin presentation then B' is strongly invertible. Such positive Artin presentation give us the fundamental group of closed, connected and orientable 3-manifolds M, and in fact, by giving an example, we show that there exist 3-manifolds whose fundamental group does not admit a positive Artin presentation.
Paper Structure (4 sections, 12 theorems, 15 equations, 11 figures)

This paper contains 4 sections, 12 theorems, 15 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Examples

Key Result

Theorem 1.1

There is a set $\mathcal{P}$ of framed pure $n$-braids, which contains all the pure $n$-braids which admit a positive Artin n-presentation (depending on the framings).

Figures (11)

  • Figure 1: The disk with $n$ holes
  • Figure 2: A Heegaard diagram given by an Artin presentation
  • Figure 3: Example with 8 chords.
  • Figure 4: The full twist along 4 chords
  • Figure 5: Diagrams of 2 holes.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 3.1
  • proof
  • ...and 14 more