Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Long--Moody construction and twisted Alexander invariants

Akihiro Takano

Abstract

In 1994, Long and Moody introduced a method to construct a new representation of the braid group from the representation of the braid group or the semidirect product of the braid group and the free group. In this paper, we show that its matrix presentation is written using the Fox derivation, and also a relation with twisted Alexander invariants.

The Long--Moody construction and twisted Alexander invariants

Abstract

In 1994, Long and Moody introduced a method to construct a new representation of the braid group from the representation of the braid group or the semidirect product of the braid group and the free group. In this paper, we show that its matrix presentation is written using the Fox derivation, and also a relation with twisted Alexander invariants.
Paper Structure (8 sections, 2 theorems, 73 equations, 9 figures)

This paper contains 8 sections, 2 theorems, 73 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Twisted Alexander invariants
  3. Colored braids
  4. The Long--Moody construction
  5. Definition
  6. Reduced version
  7. Relation with the twisted Alexander invariant
  8. Examples

Key Result

Theorem 4.2

For any $b \in B_c$,

Figures (9)

  • Figure 1: The generator $\sigma_i$
  • Figure 2: The $n$-punctured disk $D_{n}$ and the generators of its fundamental group
  • Figure 3: The action of $\sigma_i$ on $\pi_1 (D_n, z)$
  • Figure 4: The closure of a braid
  • Figure 5: $b_1$ is a $(c, c')$-braid, $b_2$ is a $(c', c")$-braid, and their composition $b_1 b_2$ is a $(c, c")$-braid.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Definition 2.1: Wada
  • Definition 4.1: cf. Long, Bigelow-Tian
  • Theorem 4.2
  • proof
  • Example 4.3
  • Remark 4.4
  • Definition 4.5
  • Example 4.6
  • Remark 4.7
  • Theorem 4.8
  • ...and 8 more