New groups and invariants of classical braids valued in $G_{n}^{2}$

TL;DR

This paper introduces new groups and invariants for classical braids using $G_{n}^{2}$ groups, simplifying the structure and solving the identity problem, thus advancing braid theory.

Contribution

It constructs novel braid invariants valued in $G_{N}^{2}$ groups and demonstrates the simplified structure and solvability of the identity problem in these groups.

Findings

Invariants valued in $G_{N}^{2}$ groups are constructed.

The structure of $G_{n}^{2}$ groups is simpler than $G_{n}^{3}$.

The identity problem in $G_{n}^{2}$ groups is solved.

Abstract

The aim of the present note is to enhance groups $G_{n}^{3}$ and to construct new invariants of classical braids. In particular, we construct invariants valued in $G_{N}^{2}$ groups. In groups $G_{n}^{2}$, the identity problem is solved, besides, their structure is much simpler than that of $G_{n}^{3}$. I am grateful to Huyue Yan for pointing out a small mistake in the previous version.