On the Burau representation modulo a small prime

$\mathbb{Z}_p$-模上的加斯纳表示与布劳表示

Gassner and Burau representations over $\mathbb{Z}_p$-modules

本文研究Artin辫群的Gassner与Burau表示及其模p约化，运用拓扑方法证明Gassner表示对所有n均忠实且模p忠实，并给出Burau表示模p忠实性的新证明，展望高辫群应用。

关于模$p$的$B_3$的Burau表示

On the Burau representation of $B_3$ modulo $p$

本研究提出判定3股辫群模素数$p$的Burau表示忠实性的算法，证明$p\le 13$时该表示忠实，还解决了Salter提出的有限域$\mathbb{F}_p$上相关问题。

On the Burau Representation of $B_4$ modulo $p$

The problem of faithfulness of the (reduced) Burau representation for $n =4$ is known to be equivalent to the problem of whether certain two matrices $A$ and $B$ generate a free group of rank two. It is known that $A^3$ and $B^3$ generate a free group of rank two \cite{9}, \cite{10}, \cite{4}. We prove that they also generate a free group when considered as matrices over the $\mathbb{Z}_p[t,t^{-1}]$ for any integer $p > 1$.

关于 $B_4$ 的 Burau 表示像的 Salter 问题

Salter's question on the image of the Burau representation of $B_4$

本文通过构造反例对 $n=4$ 情况下的 Salter 问题给出了否定解答，并提出了关于 $B_4$ Burau 表示忠实性的两点注记，包括推广了前人结果及提供建筑理论判据。

A strong characterization of the entries of the Burau matrices of $4$-braids: The Burau representation of the braid group $B_4$ is faithful almost everywhere

We establish strong constraints on the kernel of the (reduced) Burau representation $β_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitly determine the entries of the Burau matrices of braids in $B_4$, and this is an important step toward demonstrating that $β_4$ is faithful (a longstanding question posed in the 1930s). The theory is based on a novel combinatorial interpretation of $β_4\left(g\right)$, in terms of the Garside normal form of $g\in B_4$ and a new product decomposition of positive braids. We develop cance…

关于素数模下的 Jones 多项式

On the Jones polynomial modulo primes

本文研究素数模下纽结 Jones 多项式的分布密度，证明大次数范围内其上界为 $4/p^7$，并据此分类跨度至多八的模二情形。