论文标题
$ 4 $编的burau矩阵条目的强烈描述：编织组的burau代表$ b_4 $几乎是忠实的
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
论文作者
论文摘要
我们在（简化）burau表示的内核上建立了强大的约束，$β_4：b_4 \ to \ text {gl} _3 \ left（\ mathbb {z} \ left [q^{q^{\ pm 1} \ pm 1} \ right] \ right]我们开发了一种理论，可以明确确定$ b_4 $中辫子的布劳矩阵的条目，这是证明$β_4$是忠实的重要一步（1930年代提出的一个长期问题）。该理论基于$β_4\ left（g \右）$的新型组合解释，就b_4 $中的$ g \的garside正常形式和阳性辫子的新产品分解而言。我们为矩阵组中的单词开发取消结果，以表明，如果$σ$是$ b_4 $中的通用正编织，并且如果$ t \ neq 2 $是质量数字，则至少矩阵$β_4\ feft（σ\右）中的主要系数是non-Zero Modulo odulo $ $ t $。我们利用这些取消结果来推断出$ b_4 $的burau代表几乎到处都是忠实的。
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 cancellation results for words in matrix groups to show that if $σ$ is a generic positive braid in $B_4$ and if $t\neq 2$ is a prime number, then the leading coefficients in at least one row of the matrix $β_4\left(σ\right)$ are non-zero modulo $t$. We exploit these cancellation results to deduce that the Burau representation of $B_4$ is faithful almost everywhere.
