Quantum topology provides various frameworks for defining and computing invariants of manifolds. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we consider a restricted class of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. These categories are particularly simple, being entirely specified by three pieces of data: a finite abelian group, a bicharacter of that group, and a sign $\pm 1$. Despite being one of the simplest sources of state sum invariants, the computational complexities of Tambara-Yamagami invariants are yet to be fully understood. We make substantial progress on this problem. Our main result is the existence of a general fixed parameter tractable algorithm for all such topological invariants, where the parameter is the first Betti number of the 3-manifold with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We also explain that these invariants are sometimes #P-hard to compute (and we expect that this is almost always the case). Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is easily computable in polynomial time.

翻译：量子拓扑学提供了多种定义和计算流形不变量的框架。其中一个在数学和物理学领域均备受关注的框架是Turaev-Viro-Barrett-Westbury状态和构造，它利用球状融合范畴的数据，通过张量网络收缩来定义三角剖分三维流形的拓扑不变量。本文考虑一类源于Tambara-Yamagami范畴的三维流形状态和不变量的受限类别。这类范畴尤为简单，完全由三项数据指定：一个有限阿贝尔群、该群的一个双特征标，以及一个符号$\pm 1$。尽管作为状态和不变量最简单的来源之一，Tambara-Yamagami不变量的计算复杂性尚未得到完全理解。我们在该问题上取得了实质性进展。主要结果是为所有此类拓扑不变量建立了一种通用的固定参数可解算法，其中参数为三维流形在$\mathbb{Z}/2\mathbb{Z}$系数下的第一贝蒂数。我们还指出这些不变量有时属于#P-hard计算问题（并预期在绝大多数情况下均如此）。与其他计算拓扑学领域（如曲面上的图论）不同，三维流形拓扑学中已知可接受带拓扑参数的FPT算法的困难问题极少。然而，这类算法尤其引人关注，因其复杂度仅取决于输入组合表示的多项式函数，而与规模或组合宽度无关。此外，就贝蒂数而言，该参数本身可在多项式时间内轻松计算。