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: $$P{B}_{n+1}\stackrel{p_k}{\to }CP{B}_n\stackrel{f_d}{\to }VC{B}_n\stackrel{\rho }{\to }G{L}_n(\mathbb{Z}[t^{\pm 1},s^{\pm 1}])$$ In this paper, we prove that M-N is unfaithful if $k\ge 6$ since Burau kernel is a subgroup of M-N kernel.

4 pages, 2 figures