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A generalisation of the Burau representation and groups $G_{n}^{3}$ for classical braids

Vassily Olegovich Manturov, Igor Mikhailovich Nikonov

TL;DR

This paper introduces a modification of the $G^3_n$ framework by defining the group $\hat{G}^3_n$ with generators $a_{ijk}$ and specific relations, together with a $\Sigma_n$-action that yields the semidirect product $\Sigma_n\ltimes \hat{G}^3_n$. It constructs a representation $\rho$ of $\hat{G}^3_n$ on a free $A$-module $V$ and uses a braid-to-group map $\phi_n$ to obtain a representation of the pure braid group $PB_n$ via $\rho\circ\phi_n$, capable of detecting nontrivial elements in the kernel of the Burau representation. Explicit calculations show the induced representation can distinguish Burau-kernel elements (e.g., in the kernel for $PB_5$ and $PB_6$ cases), indicating it is stronger than some existing approaches. The real-form of the module, $V_\mathbb{R}$, decomposes as $V_{sym}\oplus V_{alt}$, and connections to known $V_{sym}$ representations from FKM suggest a broader, unifying representation framework for braid invariants beyond Burau.

Abstract

We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to detect the kernel of the Burau representation.

A generalisation of the Burau representation and groups $G_{n}^{3}$ for classical braids

TL;DR

This paper introduces a modification of the framework by defining the group with generators and specific relations, together with a -action that yields the semidirect product . It constructs a representation of on a free -module and uses a braid-to-group map to obtain a representation of the pure braid group via , capable of detecting nontrivial elements in the kernel of the Burau representation. Explicit calculations show the induced representation can distinguish Burau-kernel elements (e.g., in the kernel for and cases), indicating it is stronger than some existing approaches. The real-form of the module, , decomposes as , and connections to known representations from FKM suggest a broader, unifying representation framework for braid invariants beyond Burau.

Abstract

We consider a certain modification of the group which describes dynamics of point configurations, in particular braids, and define a representation of the modified . The braid representation induced is powerful enough to detect the kernel of the Burau representation.
Paper Structure (3 sections, 4 theorems, 9 equations, 2 figures)

This paper contains 3 sections, 4 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. A generalisation of the Burau representation

Key Result

Proposition 1

The map $\phi_n$ whose values in the generators $\sigma_i$, $i=1,\dots,n-1$, are induces a well defined homomorphism $\phi_n\colon B_n\to\Sigma_n\ltimes \hat{G}^3_n$.

Figures (2)

  • Figure 1: The event $a_{ijk}$
  • Figure 2: Dynamics of the generator $\sigma_i$

Theorems & Definitions (11)

  • Definition 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • proof
  • Corollary 2
  • Example 1
  • Example 2
  • ...and 1 more