Reading the dual Garside length of braids from homological and quantum representations

TL;DR

This paper demonstrates that certain homological and quantum representations of braids can directly determine the dual Garside length from their matrix spans, providing a natural and straightforward detection method.

Contribution

It introduces a novel approach linking Lawrence's and quantum sl_2 representations to the dual Garside length of braids, simplifying length detection.

Findings

Representation matrices' variable span equals dual Garside length

Homological and quantum representations detect dual Garside length

Method offers a natural way to measure braid complexity

Abstract

We show that Lawrence's representation and linear representations from quantum sl_2 called generic highest weight vectors detect the dual Garside length of braids in a simple and natural way. That is, by expressing a representation as a matrix over a Laurent polynomial ring using certain natural basis, the span of the variable is equal to the constant multiples of the dual Garside length.