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Stephan Tillmann

Publications and source records attributed to Stephan Tillmann.

At least 19 recordsLinked to original sources
arXiv2026

An Edge-Based Formulation for the Exact Computation of High-Order Zernike Moments of 2D Shapes and Images

Zernike moments are widely used rotation-invariant descriptors for shape and image analysis, but their standard computation relies on a pixel-based quadrature that treats each pixel as a point mass located at its center. This approximation introduces spatial aliasing that increases with moment order, degrading image reconstruction and reducing the discriminative power of high-order moments. We present an edge-based formulation that eliminates this source of error by applying Green's theorem to transform the two-dimensional area integral defining a Zernike moment into a sum of one-dimensional integrals along image boundaries. The resulting framework applies equally to polygonal shapes, binary images, grayscale images, and color images. We derive recurrence relations for the required radial primitives and show that the transformed edge integrands are polynomial functions, allowing their exact evaluation using Clenshaw--Curtis quadrature. The proposed method computes Zernike moments from polygonal image representations without the spatial discretization errors inherent to conventional pixel-based approaches and remains computationally practical for high-order moments. Numerical experiments on image reconstruction, shape analysis, and character classification demonstrate that the proposed formulation matches the accuracy of classical methods at low orders while remaining stable at orders for which pixel-based moments suffer from significant aliasing and numerical degradation.

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arXiv2026

Moving between 3-manifold triangulations is NP-hard

We show that \textsc{number of bistellar moves and sparse degree-two edge collapses for 3-sphere} is NP-hard. It follows that a similar problem for an arbitrary 3-manifold is NP-hard as well. This is the first NP-hardness result concerning moves between two triangulations of a 3-manifold.

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arXiv2024

Essential tori in 3-manifolds not detected in any characteristic

Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters over any algebraically closed field $\mathbb{F}$.

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arXiv2024

An invitation to Culler-Shalen theory in arbitrary characteristic

In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their $\text{SL}_2(\mathbb{C})$-character varieties, and connections between the algebraic geometry of the character variety and the topology of the 3-manifold are established via group actions on trees. Here, we lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over an arbitrary algebraically closed field. Examples include a change in the $A$-polynomial in characteristic 2, a closed Haken hyperbolic 3-manifold with no detected essential surface, and a closed Haken hyperbolic 3-manifold with an essential surface only detected in characteristic 2.

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arXiv2024

On a volume invariant of 3-manifolds

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.

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arXiv2022

Complexity of 3-manifolds obtained by Dehn filling

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.

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arXiv2022

On the topology of character varieties of once-punctured torus bundles

This paper presents, for the special case of once-punctured torus bundles, a natural method to study the character varieties of hyperbolic 3-manifolds that are bundles over the circle. The main strategy is to restrict characters to the fibre of the bundle, and to analyse the resulting branched covering map. This allows us to extend results of Steven Boyer, Erhard Luft and Xingru Zhang. Both $SL(2, \mathbb{C})$-character varieties and $PSL(2, \mathbb{C})$-character varieties are considered. As an explicit application of these methods, we build on work of Baker and Petersen to show that there is an infinite family of hyperbolic once-punctured bundles with canonical curves of $PSL(2, \mathbb{C})$-characters of unbounded genus.

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arXiv2021

A new family of minimal ideal triangulations of cusped hyperbolic 3-manifolds

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.

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arXiv2021

The Thurston norm via spun-normal immersions

A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3--manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3--manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.

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arXiv2021

Slope norm and an algorithm to compute the crosscap number

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.

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arXiv2020

The space of properly-convex structures

Suppose $G$ is finitely generated group and $\mathcal{C}(G)$ consists of all $\rho:G\to\operatorname{PGL}(n+1,\mathbb{R})$ for which there exists a properly convex set in $\mathbb{R}\mathbb{P}^n$ that is preserved by $\rho(G)$. Then the image of $\mathcal{C}(G)$ is closed in the character variety. Suppose $G$ does not contain an infinite, normal, abelian subgroup and $\mathcal{D}(G)\subset\mathcal{C}(G)$ is the subset of holonomies of properly-convex $n$-manifolds with fundamental group $G$. Then the image $\mathcal{D}(G)$ is closed in the character variety. If $M$ is the interior of a compact $n$-manifold and $G=\pi_1M$ is as above, and either $M$ is closed, or $\pi_1M$ contains a subgroup of infinite index isomorphic to $\mathbb{Z}^{n-1}$, then $\mathcal{D}(G)$ is closed. If, in addition, $M$ is the interior of a compact manifold $N$ such that every component of $\partial N$ is $\pi_1$-injective, and finitely covered by a torus, then every element of $\mathcal{D}(G)$ is the holonomy of a properly-convex structure on $M$, and $\mathcal{D}(G)$ is a union of connected components of a semi-algebraic set.

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arXiv2020

On properly convex real-projective manifolds with Generalized Cusp

Suppose $E$ is an end of an irreducible, properly convex, real-projective $n$-manifold $M$. If $\pi_1E$ contains a subgroup of finite index isomorphic to ${\mathbb Z}^{n-1}$, and $E\hookrightarrow M$ is $\pi_1$-injective, then $E$ is a generalized cusp. We list some consequences when all ends are of this type. Under certain hypotheses we prove the holonomy of a properly convex manifold is irreducible.

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arXiv2020

Projective structures on a hyperbolic 3-orbifold

We compute and analyse the moduli space of those real projective structures on a hyperbolic 3-orbifold that are modelled on a single ideal tetrahedron in projective space. Parameterisations are given in terms of classical invariants, traces, and geometric invariants, cross ratios.

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arXiv2019

On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichm\"uller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doubly decorated structures. In addition, we describe a simple, intrinsic edge-flipping algorithm to determine the canonical cell decomposition associated to a point in moduli space, and show that Penner's centres of Teichm\"uller cells are also natural centres of the cells in moduli space. We show that in many cases, the associated holonomy groups are semi-arithmetic.

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arXiv2019

Tropical varieties associated to ideal triangulations: The Whitehead link complement

This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-normal surface theory is related to the space of incompressible surfaces and to the unit ball of the Thurston norm. It is shown that all boundary curves of the Whitehead link complement are strongly detected by its character variety. The specific results obtained can be used to study the geometry and topology of the Whitehead link complement and its Dehn surgeries. The methods can be applied to any cusped hyperbolic 3--manifold of finite volume.

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arXiv2018

Computing closed essential surfaces in 3-manifolds

We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.

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arXiv2018

Traversing three-manifold triangulations and spines

A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3-manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1-vertex triangulations of closed 3-manifolds was independently proven by Matveev and Piergallini. The general result for closed 3-manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3-manifolds. These results (and their proofs) are phrased in the dual language of spines. The purpose of this note is threefold. We wish to popularise Amendola's result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions.

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