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

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.