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We reprove a necessary condition for the Sakuma-Weeks triangulation of a 2-bridge link complement to be minimal in terms of the mapping class describing its alternating 4-string braid construction. For the 2-bridge links satisfying this condition we construct explicit angle structures on the Sakuma-Weeks triangulations and compute both multiplicative and additive lower bounds on the complexity of the link complements via volume estimates. These lower bounds are an improvement on existing volume estimates for the 2-bridge links examined., Comment: 30 pages, 8 figures

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all non-hyperbolic closed 3-manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many lens spaces, the volume minimiser is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold., Comment: 26 pages, 5 figures

In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$ of a simply connected $4$-manifold $\mathcal{M}$ with $n$ pentachora, an upper bound on the number of vertices $v$ of $\mathcal{T}$ as a function of $n$ yields a lower bound for $n$ depending only on the second Betti number $\beta_2(\mathcal{M})$ of $\mathcal{M}$. Within this framework we conjecture that $v \leq \frac{n}{2}+4$, implying $n \geq 2\beta_2(\mathcal{M})$. In forthcoming work by the authors, this conjectured bound is shown to be almost tight for all values of $\beta_2(\mathcal{M})$, with a gap of at most two. We extend our conjecture to arbitrary dimensions and show that an $n$-facet triangulation of an odd-dimensional $d$-manifold, $n \geq d$, can have at most $n + \frac{d-1}{2}$ vertices, and conjecture that, for $d$ even, the bound is $\frac{n}{2}+d$. We show that these (conjectured) bounds are (would be) tight for all odd (even) dimensions and all values of $n \geq d$. Finally, we give necessary conditions for the dual graph of $\mathcal{T}$ to satisfy our conjecture. We furthermore present families of $4$-dimensional pseudomanifolds with singularities in their edge links that have more than $\frac{n}{2}+4$ vertices, thereby proving that the manifold condition is necessary for our conjecture to hold., Comment: 29 pages, 6 figures, 3 tables, 2 pages of appendix

We propose a Monte Carlo method to efficiently find, count, and sample abstract triangulations of a given manifold M. The method is based on a biased random walk through all possible triangulations of M (in the Pachner graph), constructed by combining (bi-stellar) moves with suitable chosen accept/reject probabilities (Metropolis-Hastings). Asymptotically, the method guarantees that samples of triangulations are drawn at random from a chosen probability. This enables us not only to sample (rare) triangulations of particular interest but also to estimate the (extremely small) probability of obtaining them when isomorphism types of triangulations are sampled uniformly at random. We implement our general method for surface triangulations and 1-vertex triangulations of 3-manifolds. To showcase its usefulness, we present a number of experiments: (a) we recover asymptotic growth rates for the number of isomorphism types of simplicial triangulations of the 2-dimensional sphere; (b) we experimentally observe that the growth rate for the number of isomorphism types of 1-vertex triangulations of the 3-dimensional sphere appears to be singly exponential in the number of their tetrahedra; and (c) we present experimental evidence that a randomly chosen isomorphism type of 1-vertex n-tetrahedra 3-sphere triangulation, for n tending to infinity, almost surely shows a fixed edge-degree distribution which decays exponentially for large degrees, but shows non-monotonic behaviour for small degrees., Comment: 29 pages, 6 figures

Following work by the first author and Banchoff, we investigate triangulations of real and complex projective spaces of real and complex dimension $k$ that are adapted to the decomposition into "zones of influence" around the points $[1,0,\ldots,0],$ $\ldots,$ $[0,\ldots,0,1]$ in homogeneous coordinates. The boundary of such a "zone of influence" must admit a simplicial version of the Hopf decomposition of a sphere into "solid tori" of various dimensions. We present such {\em Hopf triangulations} of $S^{2k-1}$ for $k \leq 4$, and give candidate triangulations for arbitrary $k$. In the complex case, a crucial role of this construction is the central $k$-torus as the intersection of all "zones of influence". Candidate triangulations of the $k$-torus with $2^{k+1}-1$, $k\geq 1$, vertices -- possibly the minimum numbers -- are well known. They admit an involution acting like complex conjugation and an automorphism of order $k+1$ realising the cyclic shift of coordinate directions in $\mathbb{C}P^k$. For $k=2$, this can be extended to what we call a {\em perfect equilibrium triangulation} of $\mathbb{C}P^2$, previously described in the literature. We prove that this is no longer possible for $k=3$, and no perfect equilibrium triangulation of $\mathbb{C}P^3$ exists. In the real case, the central torus is replaced by its fixed-point set under complex conjugation: the vertices of a $k$-dimensional cube. We revisit known equilibrium triangulations of $\mathbb{R}P^k$ for $k\leq 2$, and describe new equilibrium triangulations of $\mathbb{R}P^3$ and $\mathbb{R}P^4$. Finally, we discuss the most symmetric and vertex-minimal triangulation of $\mathbb{R}P^4$ and present a tight polyhedral embedding of $\mathbb{R}P^3$ into 6-space. No such embedding was known before., Comment: 31 pages, 8 figures, 5 pages of appendix

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 M yields a linear lower bound on its treewidth tw(M) (resp. pathwidth pw(M)), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of 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., Comment: 22 pages, 19 figures. The appendix has been extended

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 $\bar{g}(n)$ of our sample is very close to $\bar{g}(n) = \frac{n-1}{2} - (16.98n -110.61)^{1/4}$. 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 $n$., Comment: 12 pages, 17 figures

Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even Dehn fillings of $M$ for which our method determines the complexity of its members up to an additive constant. The constant only depends on the size of a chosen triangulation of $M$, and the isotopy class of its boundary. We then show that, given a triangulation $\mathcal T$ of $M$ with $2$--triangle torus boundary, there exist infinite families of even Dehn fillings of $M$ for which we can determine the complexity of the filled manifolds with a gap between upper and lower bound of at most $13 |\mathcal T| + 7.$ This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of $M$, or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the three-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot $P(-2,3,7)$, and the trefoil., Comment: 23 pages, 10 figures, 3 tables

Previous work of the authors with Bus Jaco determined a lower bound on the complexity of cusped hyperbolic 3-manifolds and showed that it is attained by the monodromy ideal triangulations of once-punctured torus bundles. This paper exhibits an infinite family of minimal ideal triangulations of Dehn fillings on the link $8^3_9$ that also attain this lower bound on complexity., Comment: 18 pages, 10 figures, 3 tables

We give three algorithms to determine the crosscap number of a knot in the 3-sphere using $0$-efficient triangulations and normal surface theory. Our algorithms are shown to be correct for a larger class of complements of knots in closed 3-manifolds. The crosscap number is closely related to the minimum over all spanning slopes of a more general invariant, the slope norm. For any irreducible 3-manifold $M$ with incompressible boundary a torus, we give an algorithm that, for every slope on the boundary that represents the trivial class in $H_1(M; \mathbb{Z}_2)$, determines the maximal Euler characteristic of any properly embedded surface having a boundary curve of this slope. We complement our theoretical work with an implementation of our algorithms, and compute the crosscap number of knots for which previous methods would have been inconclusive. In particular, we determine 196 previously unknown crosscap numbers in the census of all knots with up to 12 crossings., Comment: 36 pages, 14 figures, 2 tables. Improved algorithms to work in quadrilateral coordinates and included computational results

We present three "hard" diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small "standard" example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $\mathbb{S}^2$., Comment: 11 pages + appendix, 7 figures

Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p., Comment: 15 pages, 2 figures

For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable., Comment: 27 pages, 10 Figures, 4 Tables, 2 Appendices

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., Comment: 34 pages, 30 figures, 1 table

We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers., Comment: 40 pages, 5 figures. Changes from v1: new title; some proofs shortened, and some omitted. This version has been accepted for publication in The Electronic Journal of Combinatorics

Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint is now used in the study of minimal ideal triangulations. As an application, it is shown that the monodromy ideal triangulations of the hyperbolic once-punctured torus bundles are minimal., Comment: 34 pages, 20 figures, Accepted for publication by the Journal of Topology

In 1974 Chv\'atal conjectured that no intersecting family $\mathcal{F}$ in a downset can be larger than the largest star. In the same year Kleitman and Magnanti proved the conjecture when $\mathcal{F}$ is contained in the union of two stars, and Sterboul when $\operatorname{rank}(\mathcal{F})\le 3$. We give short self-contained proofs of these two statements., Comment: 3 pages, updated with additional references

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 $\mathbb{R}P^4$., Comment: 16 pages, 2 figures, 5 tables

In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol, of Scharlemann and Thompson, and of Scharlemann, Schultens and Saito by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 18(k+1) (resp. 4(3k+1))., Comment: 26 pages, 6 figures, 1 table. Apart from the formatting and updated references, this manuscript is identical to the final version published in the Journal of Computational Geometry

In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3--manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.$, Comment: 21 pages, 10 figures

We prove that the family of facets of a pure simplicial complex of dimension up to three satisfies the Erd\H{o}s-Ko-Rado property whenever it is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes., Comment: 22 pages, v2: modified title and presentation, updated references, results unchanged

The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincar\'e homology sphere)., Comment: 15 pages + 6 pages appendix, 7 figures, 4 tables

It is well-known that the Pachner graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $n\geq 4$. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph of $n$-vertex flag $2$-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph of $n$-vertex stacked $2$-spheres has at least as many connected components as there are trees on $\lfloor\frac{n-5}{3}\rfloor$ nodes with maximum node-degree at most four., Comment: 23 pages, 18 figures, 1 table

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,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 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 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 TV(4,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., Comment: 14 pages, 3 figures

For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by K\"uhnel. K\"uhnel and Lutz conjectured that any $\mathbb{F}$-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only $\mathbb{F}$-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable, neighbourly and stacked. In consequence, the K\"uhnel-Lutz conjecture is valid in dimension $\leq 3$., Comment: Corollary 1.5 is new

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 $\mathbb{Q}$., Comment: 26 pages, 10 figures, 5 tables

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimensions $\leq 3$. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension. In this paper, we present a computer-friendly combinatorial scheme to obtain tight triangulations, and present new examples in dimensions three, four and five. Furthermore, we describe a family of tight triangulated $d$-manifolds, with $2^{d-1} \lfloor d / 2 \rfloor ! \lfloor (d-1) / 2 \rfloor !$ isomorphically distinct members for each dimension $d \geq 2$. While we still do not know if there are infinitely many tight triangulations in a fixed dimension $d > 2$, this result shows that there are abundantly many., Comment: 22 pages, 3 figures, 3 tables, 13 page appendix with examples of tight triangulations available as ancillary file. New introduction, added proof of homeomorphism type of the examples, outsourced appendix

A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a sequence exist is believed to be very difficult in general. In this article we present a method, based on uniform spanning trees, to estimate how difficult it is to collapse a given $3$-sphere triangulation after removing a tetrahedron. In addition we show that out of all $3$-sphere triangulations with eight vertices or less, exactly $22$ admit a non-collapsing sequence onto a contractible non-collapsible $2$-complex. As a side product we classify all minimal triangulations of the dunce hat, and all contractible non-collapsible $2$-complexes with at most $18$ triangles. This is complemented by large scale experiments on the collapsing difficulty of $9$- and $10$-vertex spheres. Finally, we propose an easy-to-compute characterisation of $3$-sphere triangulations which experimentally exhibit a low proportion of collapsing sequences, leading to a heuristic to produce $3$-sphere triangulations with difficult combinatorial properties., Comment: 18 pages, 6 figures

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer $r \geq 3$. We resolve the question of complexity for $r=3$ and $r=4$, giving simple proofs that computing Turaev-Viro invariants for $r=3$ is polynomial time, but for $r=4$ is \#P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary $r$, and show through concrete implementation and experimentation that this algorithm is practical---and indeed preferable---to the prior state of the art for real computation., Comment: 17 pages, 5 figures

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces. In this article, we present a new polynomial time procedure to decide tightness for triangulations of $3$-manifolds -- a problem which previously was thought to be hard. Furthermore, we describe an algorithm to decide general tightness in the case of $4$-dimensional combinatorial manifolds which is fixed parameter tractable in the treewidth of the $1$-skeletons of their vertex links, and we present an algorithm to decide $\mathbb{F}_2$-tightness for weak pseudomanifolds $M$ of arbitrary but fixed dimension which is fixed parameter tractable in the treewidth of the dual graph of $M$., Comment: 18 pages, 3 figures

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $\mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $\mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $\mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $\mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $\beta_1$, then $(n-4)(617n- 3861) \leq 15444\beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra., Comment: 21 pages, 1 figure

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes., Comment: 38 pages, 25 figures

Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of "standard" type, that is, all connected sums of $\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of $\mathbb{CP}^2$ is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of $\mathbb{CP}^2$ by the minimum of four edge contractions., Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear in Advances in Geometry

This workshop about triangulations of manifolds in computational geometry and topology was held at the 2014 CG-Week in Kyoto, Japan. It focussed on computational and combinatorial questions regarding triangulations, with the goal of bringing together researchers working on various aspects of triangulations and of fostering a closer collaboration within the computational geometry and topology community. Triangulations are highly suitable for computations due to their clear combinatorial structure. As a consequence, they have been successfully employed in discrete algorithms to solve purely theoretical problems in a broad variety of mathematical research areas (knot theory, polytope theory, 2- and 3-manifold topology, geometry, and others). However, due to the large variety of applications, requirements vary from field to field and thus different types of triangulations, different tools, and different frameworks are used in different areas of research. This is why today closely related research areas are sometimes largely disjoint leaving potential reciprocal benefits unused. To address these potentials a workshop on Triangulations was held at Oberwolfach Research Institute in 2012. Since then many new collaborations between researchers of different mathematical communities have been established. Regarding the computational geometry community, the theory of manifolds continues to contribute to advances in more applied areas of the field. Many researchers are interested in fundamental mathematical research about triangulations and thus will benefit from a broad set of knowledge about different research areas using different techniques. We hope that this workshop brought together researchers from many different fields of computational geometry to have fruitful discussions which will lead to new interdisciplinary collaborations and solutions., Comment: 4 x 6 page extended abstracts. The workshop was held as part of CG-Week 2014 at Kyoto University. June 10th, 2014

We present a necessary condition for $(\ell-1)$-connected combinatorial $(2\ell +1)$-manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manifold with Betti number at most two other than the boundary of the four-simplex and the nine-vertex triangulation of the three-dimensional Klein bottle., Comment: 18 pages, 1 figure

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres $\Sigma (p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$., Comment: 28 pages, 9 figures. Minor update. To appear in Israel Journal of Mathematics

In 1987, Kalai proved that stacked spheres of dimension $d\geq 3$ are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension $d=2$. In this article, we give a characterisation of stacked $2$-spheres using what we call the {\em separation index}. Namely, we show that the separation index of a triangulated $2$-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all $n$-vertex triangulated $2$-spheres, the separation index is {\em minimised} by some $n$-vertex flag sphere for $n\geq 6$. Furthermore, we apply this characterisation of stacked $2$-spheres to settle the outstanding $3$-dimensional case of the Lutz-Sulanke-Swartz conjecture that "tight-neighbourly triangulated manifolds are tight". For dimension $d\geq 4$, the conjecture has already been proved by Effenberger following a result of Novik and Swartz., Comment: Some typos corrected, to appear in "Journal of Combinatorial Theory A"

We present new computational methods for proving diffeomorphy of triangulated 4-manifolds, including algorithms and topological software that can for the first time effectively handle the complexities that arise in dimension four and be used for large scale experiments., Comment: 4 pages, presented at the Young Researchers Forum at SCG 2013

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. 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 investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds., Comment: To appear in Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry (SoCG). 25 pages, 8 figures, 2 tables

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., Comment: A 10-page extended abstract of this work will appear in ALENEX 2013; this is the full version of the paper including details of proofs. 23 pages, 10 figures