Knot theory is an active field of mathematics, in which combinatorial and computational methods play an important role. One side of computational knot theory, that has gained interest in recent years, both for complexity analysis and practical algorithms, is quantum topology and the computation of topological invariants issued from the theory. In this article, we leverage the rigidity brought by the representation-theoretic origins of the quantum invariants for algorithmic purposes. We do so by exploiting braids and the algebraic properties of the braid group to describe, analyze, and implement a fast algorithm to compute the Hecke representation of the braid group. We apply this construction to design a parameterized algorithm to compute the HOMFLY-PT polynomial of knots, and demonstrate its interest experimentally. Finally, we combine our fast Hecke representation algorithm with Garside theory, to implement a reservoir sampling search and find non-trivial braids with trivial Hecke representations with coefficients in $\mathbb{Z}/p\mathbb{Z}$. We find several such braids, in particular proving that the Hecke representation of $B_5$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients is non-faithful, a previously unknown fact.

翻译：纽结理论是数学中的一个活跃领域，其中组合与计算方法发挥着重要作用。近年来，量子拓扑学及其衍生的拓扑不变量计算在计算纽结理论中备受关注，这既涉及复杂性分析，也涉及实用算法设计。本文利用量子不变量源于表示论的刚性特性，以实现算法优化。具体而言，我们通过辫子及辫群的代数性质，描述、分析并实现了一种计算辫群Hecke表示的快速算法。基于此构造，我们设计了一种参数化算法来计算纽结的HOMFLY-PT多项式，并通过实验验证了其价值。最后，我们将快速Hecke表示算法与Garside理论相结合，实现了蓄水池抽样搜索，以寻找在$\mathbb{Z}/p\mathbb{Z}$系数下具有平凡Hecke表示的非平凡辫子。我们发现了若干此类辫子，特别证明了$B_5$在$\mathbb{Z}/2\mathbb{Z}$系数下的Hecke表示是非忠实的，这是一个先前未知的结论。