In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants \(\mathrm {TV}_{4,q}\), using the first Betti number, i.e. the dimension of the first homology group of the manifold with \(\mathbb {Z}_2\)-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of \(\mathrm {TV}_{4,q}\) is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first \(\mathbb {Z}_2\)-homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies \(\mathrm {TV}_{4,q}\) to be added to the short list of standard properties (such as orientability, connectedness and Betti numbers) that can be computed ad hoc when first investigating an unknown triangulation.

在本文中，我们介绍了使用第一个Betti数（即，第一个同源群的维数）来计算Turaev-Viro不变量\（\ mathrm {TV} _ {4，q} \）的固定参数可处理算法。以\（\ mathbb {Z} _2 \）-系数作为参数的流形。据我们所知，这是使用拓扑参数的计算3流形拓扑中的第一个参数化算法。通常，\（\ mathrm {TV} _ {4，q} \）的计算是＃P-hard；使用拓扑参数，可为第一个\（\ mathbb {Z} _2 \）的3个流形族提供输入三角剖分大小的算法多项式-有界维的同调群。我们的算法易于实现，运行时间与运行时间相当，可为三角三歧管的标准库计算积分同源组。我们可以用这种方式计算的不变性是强大的：结合整数同源性和使用标准数据集，我们几乎可以将可以区分的3个歧管对加倍。我们希望这有资格将\（\ mathrm {TV} _ {4，q} \）添加到标准属性的简短列表（例如定向性，连通性和Betti数）中，这些标准属性可以在首次调查未知对象时临时计算三角剖分。