论文标题
$ \ mathbb {z} _p $ -modules上的gassner和burau表示
Gassner and Burau representations over $\mathbb{Z}_p$-modules
论文作者
论文摘要
我们研究了Artin的编织组及其模型$ p $降低的两个经典表示。我们使用拓扑方法表明gassner表示$τ_n：b_n \ to \ text {gl} _n（\ mathbb {z}} [t_1^{\ pm 1}，\ ldots，\ ldots，t_n^{\ pm 1}] $ P> 1 $。然后，我们给出一个新颖的证据，表明$ b_3 $的burau代表是所有$ p> 1 $的忠实Modulo $ p $，并建议向Modulo $ p $ burau代表申请较高的编织组。
We study two classical representations of Artin's braid group and their modulo $p$ reductions. We use topological methods to show that the Gassner representation $τ_n: B_n\to\text{GL}_n(\mathbb{Z}[t_1^{\pm 1}, \ldots, t_n^{\pm 1}])$ is faithful for all $n$, and furthermore that it is faithful modulo $p$ for all integers $p>1$. We then give a novel proof that the Burau representation of $B_3$ is faithful modulo $p$ for all $p>1$, and suggest applications to the modulo $p$ Burau representation for higher braid groups.
