Bowling ball representations of braid groups

TL;DR

This paper extends Jones's bowling ball metaphor for braid groups to multiple balls, leading to new algebraic representations including the Iwahori-Hecke algebra and a cabled Temperley-Lieb version.

Contribution

It introduces a multi-ball extension of the bowling ball model, resulting in novel algebraic representations of braid groups.

Findings

Derived the Iwahori-Hecke algebra from the extended model

Constructed a cabled Temperley-Lieb representation

Enhanced understanding of braid group representations

Abstract

In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain the Iwahori-Hecke algebra and a cabled version of the Temperley-Lieb representation.