Complexity of relations in the braid group

TL;DR

This paper demonstrates that in the braid group, certain identity-representing words require a quadratic number of specific relations to reduce to the empty word, highlighting the complexity of relations.

Contribution

It establishes the existence of words in braid groups where the number of braid relations needed for reduction grows quadratically with word length.

Findings

Existence of words with quadratic relation complexity

Relation count grows quadratically with word length

Highlights complexity in braid group relations

Abstract

We show that for any given n, there exists a sequence of words a_k in the generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the identity element of B_n, such that the number of braid relations of the form sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} needed to pass from a_k to the empty word is quadratic with respect to the length of a_k.