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

Yangzhou Liu

Abstract

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

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: In this paper, we prove that M-N is unfaithful if since Burau kernel is a subgroup of M-N kernel.

Paper Structure

This paper contains 3 sections, 2 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Unfaithfulness of M-N map

Key Result

Theorem 1

Manturov-Nikonov map $\rho\circ f_d\circ p_k$ is unfaithful for $k\geq 6$ and any $d\in \mathbb{Z}$.

Figures (2)

  • Figure 1: $p_k$
  • Figure 2: $f_d$

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof