A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants TV4,q of 3-Manifolds with Bounded First Betti Number

In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants TV 4 , q , using the first Betti number, i.e. the dimension of the first homology group of the manifold with 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 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 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 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.