Abstract According to a well-known result in geometric topology, we have $$\left( \mathbb {S}^2 \right) ^{n}\!\!/\operatorname {Sym}(n) = \mathbb{C}\mathbb{P}^{n}$$ S 2 n / Sym ( n ) = C P n , where $$\operatorname {Sym}(n)$$ Sym ( n ) acts on $$\left( \mathbb {S}^2 \right) ^{n}$$ S 2 n by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $$\mathbb{C}\mathbb{P}^{n}$$ C P n for each $$n \ge 2$$ n ≥ 2 . In more detail, we start with the standard two triangle crystallisation $$S^2_3$$ S 3 2 of the 2-sphere $$\mathbb {S}^2$$ S 2 , in its n -fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the $$\operatorname {Sym}(n)$$ Sym ( n ) quotient of this subdivision yields a simplicial cell decomposition of $$\mathbb{C}\mathbb{P}^n$$ C P n . The first derived subdivision of this cell complex is a simplicial triangulation of $$\mathbb{C}\mathbb{P}^n$$ C P n . To the best of our knowledge, this is the first explicit description of triangulations of $$\mathbb{C}\mathbb{P}^n$$ C P n for $$n \ge 4.$$

We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the 19-crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in 3-manifold topology.

We present a framework to classify PL-types of large censuses of triangulated $4$-manifolds, which we use to classify the PL-types of all triangulated $4$-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the $4$-sphere, $\mathbb{C}P^2$, and the rational homology sphere $QS^4(2)$, where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations -- which we deem interesting in their own rights.

Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister moves, one sometimes needs to insert excess crossings, there are significant gaps between the best known lower and upper bounds on the required number of these added crossings. In this article, we study the problem of turning a diagram of a split link into a split diagram, and we show that there exist split links with diagrams requiring an arbitrarily large number of such additional crossings. More precisely, we provide a family of diagrams of split links, so that any sequence of Reidemeister moves transforming a diagram with c crossings into a split diagram requires going through a diagram with Ω(√c) extra crossings. Our proof relies on the framework of bubble tangles, as introduced by the first two authors, and a technique of Chambers and Liokumovitch to turn homotopies into isotopies in the context of Riemannian geometry.

Abstract Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we show that a “sufficiently complicated” JSJ decomposition of a 3-manifold enforces a “complicated structure” for all of its triangulations. More concretely, we show that, under certain conditions, the treewidth (resp. pathwidth) of the graph that captures the incidences between the pieces of the JSJ decomposition of an irreducible, closed, orientable 3-manifold $$\mathscr {M}$$ M yields a linear lower bound on its treewidth $$\operatorname {tw} (\mathscr {M})$$ tw ( M ) (resp. pathwidth $$\operatorname {pw} (\mathscr {M})$$ pw ( M ) ), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of $$\mathscr {M}$$ M . We present several applications of this result. We give the first example of an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth. We construct Haken 3-manifolds with arbitrarily large treewidth—previously the existence of such 3-manifolds was only known in the non-Haken case. We also show that the problem of providing a constant-factor approximation for the treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to computing a constant-factor approximation for the treewidth (resp. pathwidth) of 3-manifolds.

We present a procedure to sample uniformly from the set of combinatorial isomorphism types of balanced triangulations of surfaces — also known as graph-encoded surfaces. For a given number n, the sample is a weighted set of graph-encoded surfaces with 2n triangles. The sampling procedure relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group. We implement our method and present a number of experimental findings based on the analysis of 138 million runs of our sampling procedure, producing graph-encoded surfaces with up to 280 triangles. Namely, we determine that, for n fixed, the empirical mean genus of our sample is very close to . Moreover, we present experimental evidence that the associated genus distribution more and more concentrates on a vanishing portion of all possible genera as n tends to infinity. Finally, we observe from our data that the mean number of non-trivial symmetries of a uniformly chosen graph encoding of a surface decays to zero at a rate super-exponential in n.

Full version of the conference paper. 22 pages, 19 figures

Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold $M$ defined by \[ \operatorname{tw}(M) = \min\{\operatorname{tw}(\Gamma(\mathcal{T})):\mathcal{T}~\text{is a triangulation of }M\}, \] where $\Gamma(\mathcal{T})$ denotes the dual graph of $\mathcal{T}$. In this setting the relationship between the topology of a 3-manifold and its treewidth is of particular interest. First, as a corollary of work of Jaco and Rubinstein, we prove that for any closed, orientable 3-manifold $M$ the treewidth $\operatorname{tw}(M)$ is at most $4\mathfrak{g}(M)-2$ where $\mathfrak{g}(M)$ denotes the Heegaard genus of $M$. In combination with our earlier work with Wagner, this yields that for non-Haken manifolds the Heegaard genus and the treewidth are within a constant factor. Second, we characterize all 3-manifolds of treewidth one: These are precisely the lens spaces and a single other Seifert fibered space. Furthermore, we show that all remaining orientable Seifert fibered spaces over the 2-sphere or a non-orientable surface have treewidth two. In particular, for every spherical 3-manifold we exhibit a triangulation of treewidth at most two. Our results further validate the parameter of treewidth (and other related parameters such as cutwidth, or congestion) to be useful for topological computing, and also shed more light on the scope of existing FPT algorithms in the field.

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4-manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the simplicial structure of triangulations using the Budney-Burton census of closed triangulated 4-manifolds. These experiments include the construction of minimal genus trisections of the non-orientable 4-manifolds $S^3 \tilde{\times} S^1$ and $\mathbb{R}P^4$.

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV4,q, using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of TV4,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 extremely large family of 3-manifolds with first homology group of bounded rank. 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 roughly double the pairs of 3-manifolds we can distinguish. We hope this qualifies TV4,q to be added to the short list of standard properties (such as orientability, connectedness, Betti numbers, etc.) that can be computed ad-hoc when first investigating an unknown triangulation.

Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpreted as embeddings of surfaces in T. We give a characterisation of how these embedded surfaces intersect with the tetrahedra of T. This is done by characterising isotopy classes of simple closed loops in the 3-punctured disk. As a direct result we obtain a new system of coordinates for edge colourings which allows for simpler definitions of the tetrahedron weights incorporated in the Turaev-Viro invariants. Moreover, building on a detailed analysis of the colourings, as well as classical work due to Kirby and Melvin, Matveev, and others, we show that considering a much smaller set of colourings suffices to compute Turaev-Viro invariants in certain significant cases. This results in a substantial improvement of running times to compute the invariants, reducing the number of colourings to consider by a factor of $2^n$. In addition, we present an algorithm to compute Turaev-Viro invariants of degree four -- a problem known to be #P-hard -- which capitalises on the combinatorial structure of the input. The improved algorithms are shown to be optimal in the following sense: There exist triangulations admitting all colourings the algorithms consider. Furthermore, we demonstrate that our new algorithms to compute Turaev-Viro invariants are able to distinguish the majority of $\mathbb{Z}$-homology spheres with complexity up to $11$ in $O(2^n)$ operations in $\mathbb{Q}$.

Optimal Morse matchings reveal essential structures of cell complexes that lead to powerful tools to study discrete geometrical objects, in particular, discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand, we prove that the erasability problem is W [ P ]-complete on the natural parameter. On the other hand, we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds, which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplices. This algorithm also shows fixed-parameter tractability for problems such as erasability and maximum alternating cycle-free matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we discuss the topological significance of the chosen parameters and investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.

In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance. Although this theory typically leads to exponential time algorithms, very little is known about how these algorithms perform in “typical” scenarios, or how far the best known theoretical bounds are from the real worst-case scenarios. Here we study the combinatorial and algebraic complexity of normal surfaces from both the theoretical and experimental viewpoints. Theoretically, we obtain new exponential lower bounds on the worst-case complexities in a variety of settings that are important for practical computation. Experimentally, we study the worst-case and average-case complexities over a comprehensive body of roughly three billion input triangulations. Many of our lower bounds are the first known exponential lower bounds in these settings, and experimental evidence suggests that many of our theoretical lower bounds on worst-case growth rates may indeed be asymptotically tight.

Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W[P]-complete on the natural parameter. On the other hand we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplexes. This algorithm also shows fixed parameter tractability for problems such as erasability and maximum alternating cycle-free matching.

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting taut angle structures is NP-complete, but also fixed-parameter tractable in the treewidth of the face pairing graph of the triangulation. These results have deeper implications: the core techniques can serve as a launching point for approaching key decision problems such as unknot recognition and prime decomposition of 3-manifolds.