The nontrivial kernel of Manturov-Nikonov map from classical braids to virtual braids

TL;DR

This paper investigates the kernel of a composite map from classical to virtual braids, showing it is nontrivial and unfaithful for certain parameters, with implications for braid group representations.

Contribution

It proves the unfaithfulness of the Manturov-Nikonov map for specific cases, revealing the Burau kernel as a subgroup of its kernel.

Findings

The Manturov-Nikonov map is unfaithful if k ≥ 6.

The Burau kernel is contained within the kernel of this map.

The result impacts the understanding of braid group representations.

Abstract

In 2022, V. O. Manturov and I. M. Nikonov \cite{Man22} constructed two composite maps, one of them is following: $$PB_{n+1} \overset{p_k}{\rightarrow} CPB_{n} \overset{f_d}{\rightarrow} VCB_{n}\overset{\rho}{\rightarrow} GL_n(\mathbb{Z}[t^{\pm 1},s^{\pm 1}])$$ In this paper, we prove that M-N is unfaithful if $k\geq 6$ since Burau kernel is a subgroup of M-N kernel.