Group G_{n}^{3} and imaginary generators

TL;DR

This paper introduces a new invariant for classical braids by embedding the pure braid group into a larger group with additional generators, enabling the identification of minimal braid words.

Contribution

It constructs a monomorphism from the pure braid group into a larger group with new generators, creating a novel invariant for classical braids.

Findings

New invariant distinguishes minimal braid words

Embedding provides a method to analyze braid minimality

Examples demonstrate the invariant's effectiveness

Abstract

In the present paper, we construct a monomorphism from (Artin) pure braid group $PB_{n}$ into a group, which is `bigger' than $PB_{n}$. Roughly speaking, this mapping is defined on words of braids by adding `new generators' between generators of $PB_{n}$. By this mapping we can get a new invariant for classical braids. As one of application of this invariant, we will show examples, which are minimal words in $PB_{n}$ and the minimality can be shown by the invariant.