arXiv subjects

Jonathan Spreer

Publications and source records attributed to Jonathan Spreer.

At least 55 records · Page 3Linked to original sources

arXiv2016

A characterization of tightly triangulated 3-manifolds

For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by Kühnel. Kühnel 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 Kühnel-Lutz conjecture is valid in dimension $\leq 3$.

math.GT↗

arXiv2016

Tight triangulations of closed 3-manifolds

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 Kühnel-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 $β_1$, then $(n-4)(617n- 3861) \leq 15444β_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.

math.GT↗

arXiv2015

Admissible colourings of 3-manifold triangulations for Turaev-Viro type invariants

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 $\mathbb{Q}$.

cs.CG↗

arXiv2015

Random collapsibility and 3-sphere recognition

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.

math.GT↗

arXiv2015

Combinatorial 3-manifolds with transitive cyclic symmetry

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.

math.CO↗

arXiv2015

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

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 $Σ(p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$.

math.GT↗

arXiv2015

Partitioning the triangles of the cross polytope into surfaces

We present a constructive proof that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope $β^k$ into closed surfaces of genus $g \leq 1$, each with a transitive automorphism group given by the vertex transitive $\mathbb{Z}_{2k}$-action on $β^k$. Furthermore we show that for each $k \equiv 1,5(6)$ the 2-skeleton of the (k-1)-simplex is a union of highly symmetric tori and Möbius strips.

math.CO↗

arXiv2015

Separation index of graphs and stacked 2-spheres

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.

math.GT↗

arXiv2015

Simple crystallizations of 4-manifolds

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.

math.GT↗

arXiv2015

Algorithms and complexity for Turaev-Viro invariants

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.

math.GT↗

arXiv2014

Efficient algorithms to decide tightness

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 $M$.

cs.CG↗

arXiv2014

Bounds for the genus of a normal surface

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.

math.GT↗

arXiv2014

Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto

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.

cs.CG↗

arXiv2014

A necessary condition for the tightness of odd-dimensional combinatorial manifolds

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.

math.CO↗

arXiv2014

Computationally proving triangulated 4-manifolds to be diffeomorphic

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.

math.GT↗

arXiv2013

Parameterized Complexity of Discrete Morse Theory

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.

cs.CG↗

arXiv2012

Computational topology and normal surfaces: Theoretical and experimental complexity bounds

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.

math.GT↗

arXiv2012

The complexity of detecting taut angle structures on triangulations

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 decision problems such as unknot recognition and prime decomposition of 3-manifolds.

math.GT↗