Publications (58)
The search for exotic knot traces
Marc Kegel, Jonathan Spreer
Two distinct knots are said to be friends if their complements, filled along the 0-slope, produce diffeomorphic 3-manifolds. In this article, we develop a practical algorithm, impl…
Triangulating Spun 2-Knot Complements
Rhuaidi Antonio Burke, Benjamin Burton, Arunima Ray +3
A -knot is an embedding of a -sphere into the -sphere. Similar to the case of embedding circles into the -sphere, this allows the -sphere to be knotted. In this shor…
Parameterized Complexity of Discrete Morse Theory
Benjamin A. Burton, Thomas Lewiner, João Paixão +1
Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. Howeve…
Parametrized Complexity of Expansion Height
Ulrich Bauer, Abhishek Rathod, Jonathan Spreer
Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of th…
A polynomial time algorithm to compute quantum invariants of 3-manifolds with bounded first Betti number
Clément Maria, Jonathan Spreer
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 manifol…
Combinatorial Seifert fibred spaces with transitive cyclic automorphism group
Benjamin Burton, Jonathan Spreer
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…
Efficient algorithms to decide tightness
Bhaskar Bagchi, Benjamin A. Burton, Basudeb Datta +2
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…
The Pachner graph of 2-spheres
Benjamin A. Burton, Basudeb Datta, Jonathan Spreer
It is well-known that the Pachner graph of -vertex triangulated -spheres is connected, i.e., each pair of -vertex triangulated -spheres can be turned into each other by…
A necessary condition for the tightness of odd-dimensional combinatorial manifolds
Jonathan Spreer
We present a necessary condition for -connected combinatorial -manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manif…
Slope norm and an algorithm to compute the crosscap number
William Jaco, J. Hyam Rubinstein, Jonathan Spreer +1
We give three algorithms to determine the crosscap number of a knot in the 3-sphere using -efficient triangulations and normal surface theory. Our algorithms are shown to be cor…
On the treewidth of triangulated 3-manifolds
Kristóf Huszár, Jonathan Spreer, Uli Wagner
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, s…
Z2-Thurston Norm and Complexity of 3-Manifolds, II
William Jaco, J. Hyam Rubinstein, Jonathan Spreer +1
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…
Small Triangulations of -Manifolds: Introducing the -Manifold Census
Rhuaidi Antonio Burke, Benjamin A. Burton, Jonathan Spreer
We present a framework to classify PL-types of large censuses of triangulated -manifolds, which we use to classify the PL-types of all triangulated -manifolds with up to six…
Independence complexes of circle graphs
Rhea Palak Bakshi, Ali Guo, Dionne Ibarra +4
Independence complexes of circle graphs are purely combinatorial objects. However, when constructed from some diagram of a link , they reveal topological properties of , more…
Short proof of two cases of Chvátal's conjecture
Jorge Olarte, Francisco Santos, Jonathan Spreer
In 1974 Chvátal conjectured that no intersecting family in a downset can be larger than the largest star. In the same year Kleitman and Magnanti proved the conjectur…
Unravelling the Dodecahedral Spaces
Jonathan Spreer, Stephan Tillmann
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 hyperbol…
A census of face-transitive surfaces
Reymond Akpanya, Jonathan Spreer
A face-transitive surface is a triangulated 2-dimensional manifold whose automorphism group acts transitively on its set of triangles. In this paper, we investigate this class of h…
On the width of complicated JSJ decompositions
Kristóf Huszár, Jonathan Spreer
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. Bu…
Random collapsibility and 3-sphere recognition
João Paixão, Jonathan Spreer
A triangulation of a -manifold can be shown to be homeomorphic to the -sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence…
Admissible colourings of 3-manifold triangulations for Turaev-Viro type invariants
Clément Maria, Jonathan Spreer
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invarian…
Computational topology and normal surfaces: Theoretical and experimental complexity bounds
Benjamin A. Burton, João Paixão, Jonathan Spreer
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 expone…
Hopf triangulations of spheres and equilibrium triangulations of projective spaces
Wolfgang Kühnel, Jonathan Spreer
Following work by the first author and Banchoff, we investigate triangulations of real and complex projective spaces of real and complex dimension that are adapted to the decom…
On minimal ideal triangulations of cusped hyperbolic 3-manifolds
William Jaco, Hyam Rubinstein, Jonathan Spreer +1
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-dime…
Complexity of 3-manifolds obtained by Dehn filling
William Jaco, J. Hyam Rubinstein, Jonathan Spreer +1
Let 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 As an app…
Vertex Bounds in Triangulated -Manifolds and an Application to 4-Manifold Complexity
Jonathan Spreer, Lucy Tobin
We investigate face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of connections between the combinatorics of triangula…
On the complexity of 2-bridge link complements
James Morgan, Jonathan Spreer
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 br…
Triangulations of the 3-sphere with knotted edge
Dionne Ibarra, Daniel V. Mathews, Jessica S. Purcell +1
We prove that for any knot , there exists a one-vertex triangulation of the -sphere containing an edge forming . The proof is constructive, and based on fully augmented li…
A construction principle for tight and minimal triangulations of manifolds
Benjamin A. Burton, Basudeb Datta, Nitin Singh +1
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…
Combinatorial properties of the K3 surface: Simplicial blowups and slicings
Jonathan Spreer, Wolfgang Kühnel
The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal reso…
Bounds for the genus of a normal surface
William Jaco, Jesse Johnson, Jonathan Spreer +1
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---d…
On a volume invariant of 3-manifolds
Marc Kegel, Arunima Ray, Jonathan Spreer +2
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 complemen…
Average Betti numbers of induced subcomplexes in triangulations of manifolds
Giulia Codenotti, Francisco Santos, Jonathan Spreer
We study a variation of Bagchi and Datta's -vector of a simplicial complex , whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of .…
Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes
Joseph Doolittle, Jean-Philippe Labbé, Carsten E. M. C. Lange +3
For -dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hype…
Computationally proving triangulated 4-manifolds to be diffeomorphic
Benjamin A. Burton, Jonathan Spreer
We present new computational methods for proving diffeomorphy of triangulated 4-manifolds, including algorithms and topological software that can for the first time effectively han…
Normal surfaces as combinatorial slicings
Jonathan Spreer
We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case…
Combinatorial 3-manifolds with transitive cyclic symmetry
Jonathan Spreer
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 w…
A Practical Algorithm for Knot Factorisation
Alexander He, Eric Sedgwick, Jonathan Spreer
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 inpu…
A characterization of tightly triangulated 3-manifolds
Bhaskar Bagchi, Basudeb Datta, Jonathan Spreer
For a field , the notion of -tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that any -tight triangula…
Sampling triangulations of manifolds using Monte Carlo methods
Eduardo G. Altmann, Jonathan Spreer
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 possib…
Partitioning the triangles of the cross polytope into surfaces
Jonathan Spreer
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 , each with a…
Tight triangulations of closed 3-manifolds
Bhaskar Bagchi, Basudeb Datta, Jonathan Spreer
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 t…
Small Triangulations of Simply Connected 4-Manifolds
Jonathan Spreer, Lucy Tobin
We present small triangulations of all connected sums of and with the standard piecewise linear structure. Our triangulations have pentac…
A new family of minimal ideal triangulations of cusped hyperbolic 3-manifolds
J. Hyam Rubinstein, Jonathan Spreer, Stephan Tillmann
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 triangu…
Hard diagrams of split links
Corentin Lunel, Arnaud de Mesmay, Jonathan Spreer
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…
3-Manifold triangulations with small treewidth
Kristóf Huszár, Jonathan Spreer
Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold defined by \[ \operatorname{tw}(M)…
Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations
Jonathan Spreer, Stephan Tillmann
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 genu…
Simple crystallizations of 4-manifolds
Biplab Basak, Jonathan Spreer
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 ad…
Algorithms and complexity for Turaev-Viro invariants
Benjamin A. Burton, Clément Maria, Jonathan Spreer
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but curr…
The complexity of detecting taut angle structures on triangulations
Benjamin A. Burton, Jonathan Spreer
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-manif…
simpcomp -- A GAP toolbox for simplicial complexes
Felix Effenberger, Jonathan Spreer
simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abs…
Simplicial cell decompositions of
Basudeb Datta, Jonathan Spreer
According to a well-known result in geometric topology, we have \linebreak , where $\operatorname{Sym}(…
Supplemental material to the article "Partitions of the triangles of the cross polytope into surfaces''
Jonathan Spreer
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 \leq 1, each with a tran…
Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto
Jonathan Spreer, Uli Wagner, Benjamin A. Burton +3
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…
A Uniform Sampling Procedure for Abstract Triangulations of Surfaces
Rajan Shankar, Jonathan Spreer
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 give…
Simplicial blowups and discrete normal surfaces in simpcomp
Felix Effenberger, Jonathan Spreer
simpcomp is an extension to GAP, the well known system for computational discrete algebra. It allows the user to work with simplicial complexes. In the latest version, support for…
Hard Diagrams of the Unknot
Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler +5
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…
The EKR property for flag pure simplicial complexes without boundary
Jorge Olarte, Francisco Santos, Jonathan Spreer +1
We prove that the family of facets of a pure simplicial complex of dimension up to three satisfies the ErdÅs-Ko-Rado property whenever it is flag and has no boundary ridges. We co…
Separation index of graphs and stacked 2-spheres
Benjamin A. Burton, Basudeb Datta, Nitin Singh +1
In 1987, Kalai proved that stacked spheres of dimension are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result…