arXiv subjects

J. Hyam Rubinstein

Publications and source records attributed to J. Hyam Rubinstein.

At least 19 recordsLinked to original sources

arXiv2025

The 3D index and Dehn filling

We provide a rigorous proof of the Gang-Yonekura formula describing the transformation of the 3D index under Dehn filling a cusp in an orientable 3-manifold. The 3D index, originally introduced by Dimofte, Gaiotto and Gukov, is a physically inspired q-series that encodes deep topological and geometric information about cusped 3-manifolds. Building on the interpretation of the 3D index as a generating function over Q-normal surfaces, we introduce a relative version of the index for ideal triangulations with exposed boundary. This notion allows us to formulate a relative Gang-Yonekura formula, which we prove by developing a gluing principle for relative indices and establishing an inductive framework in the case of layered solid tori. Our approach makes use of Garoufalidis-Kashaev's meromorphic extension of the index, along with new identities involving q-hypergeometric functions. As an application, we study the limiting behaviour of the index for large fillings. We also develop code to perform certified computations of the index, guaranteeing correctness up to a specified accuracy. Our extensive computations support the topological invariance of the 3D index and suggest a well-defined extension to closed manifolds.

math.GT↗

arXiv2025

Compression Maps Between Polytopes

A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak compression), meaning a distance decreasing (respectively a distance non-increasing) map between distinct pairs of points. We establish that there is a partial order on the space of shapes given by the relation of having a weak compression map between pairs of shapes. Finally, we construct a compression metric on the projective shape space of a polytope; the space of convex Euclidean realizations modulo rigid motions and homothety. For the projective space shape of a simplex, we show that the compression metric is complete. For general polytopes, we establish that the projective shape space has a natural completion given by the projective shape space of weakly convex realizations.

math.MG↗

arXiv2022

Unlabelled Sample Compression Schemes for Intersection-Closed Classes and Extremal Classes

The sample compressibility of concept classes plays an important role in learning theory, as a sufficient condition for PAC learnability, and more recently as an avenue for robust generalisation in adaptive data analysis. Whether compression schemes of size $O(d)$ must necessarily exist for all classes of VC dimension $d$ is unknown, but conjectured to be true by Warmuth. Recently Chalopin, Chepoi, Moran, and Warmuth (2018) gave a beautiful unlabelled sample compression scheme of size VC dimension for all maximum classes: classes that meet the Sauer-Shelah-Perles Lemma with equality. They also offered a counterexample to compression schemes based on a promising approach known as corner peeling. In this paper we simplify and extend their proof technique to deal with so-called extremal classes of VC dimension $d$ which contain maximum classes of VC dimension $d-1$. A criterion is given which would imply that all extremal classes admit unlabelled compression schemes of size $d$. We also prove that all intersection-closed classes with VC dimension $d$ admit unlabelled compression schemes of size at most $11d$.

cs.LG↗

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.

math.GT↗

arXiv2022

Counting essential surfaces in 3-manifolds

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give algorithms that allow us to compute these generating functions and the underlying surfaces, and apply these to almost 60,000 manifolds, providing a wealth of data about them. We use this data to explore the delicate question of counting only connected essential surfaces and propose some conjectures. Our methods involve normal and almost normal surfaces, especially the work of Tollefson and Oertel, combined with techniques pioneered by Ehrhart for counting lattice points in polyhedra with rational vertices. We also introduce a new way of testing if a normal surface in an ideal triangulation is essential that avoids cutting the manifold open along the surface; rather, we use almost normal surfaces in the original triangulation.

math.GT↗

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.

math.GT↗

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.

math.GT↗

arXiv2021

A Gradient Descent Method for The Dubins Traveling Salesman Problem

We propose a combination of a bounding procedure and gradient descent method for solving the Dubins traveling salesman problem, that is, the problem of finding a shortest curvature-constrained tour through a finite number of points in the euclidean plane. The problem finds applications in path planning for robotic vehicles and unmanned aerial vehicles, where a minimum turning radius prevents the vehicle from taking sharp turns. In this paper, we focus on the case where any two points are separated by at least four times the minimum turning radius, which is most interesting from a practical standpoint. The bounding procedure efficiently determines the optimal order in which to visit the points. The gradient descent method, which is inspired by a mechanical model, determines the optimal trajectories of the tour through the points in a given order, and its computation time scales linearly with the number of points. In experiments on nine points, the bounding procedure typically explores no more than a few sequences before finding the optimal sequence, and the gradient descent method typically converges to within 1\% of optimal in a single iteration.

math.OC↗

arXiv2020

Efficient triangulations and boundary slopes

For a compact, irreducible, $\partial$-irreducible, an-annular bounded 3-manifold $M\ne\mathbb{B}^3$, then any triangulation $\mathcal{T}$ of $M$ can be modified to an ideal triangulation $\mathcal{T}^*$ of $\stackrel{\circ}{M}$. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold $M$ admitting an annular-efficient triangulation are sufficient to modify any triangulation of $M$ to a boundary-efficient triangulation which is also annular-efficient. From the proof we have for any ideal triangulation $T^*$ and any inflation $\mathcal{T}_Λ$, there is a bijective correspondence between the closed normal surfaces in $\mathcal{T}^*$ and the closed normal surfaces in $\mathcal{T}_Λ$ with corresponding normal surfaces being homeomorphic. It follows that for an ideal triangulation $\mathcal{T}^*$ that is $0$-efficient, $1$-efficient, or end-efficient, then any inflation $\mathcal{T}_Λ$ of $\mathcal{T}^*$ is $0$-efficient, $1$-efficient, or $\partial$-efficient, respectively. There are algorithms to decide if a given triangulation or ideal triangulation of a $3$-manifold is one of these efficient triangulations. Finally, it is shown that for an annular-efficient triangulation, there are only a finite number of boundary slopes for normal surfaces of a bounded Euler characteristic; hence, in a compact, orientable, irreducible, $\partial$-irreducible, and an-annular $3$-manifold, there are only finitely many boundary slopes for incompressible and $\partial$-incompressible surfaces of a bounded Euler characteristic.

math.GT↗

arXiv2020

Immersed flat ribbon knots

We study the minimum ribbonlength for immersed planar ribbon knots and links. Our approach is to embed the space of such knots and links into a larger more tractable space of disk diagrams. When length minimisers in disk diagram space are ribbon, then these solve the ribbonlength problem. We also provide examples when minimisers in the space of disk diagrams are not ribbon and state some conjectures. We compute the minimal ribbonlength of some small knot and link diagrams and certain infinite families of link diagrams. Finally we present a bound for the number of crossings for a diagram yielding the minimum ribbonlength of a knot or link amongst all diagrams.

math.GT↗

arXiv2019

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.

math.GT↗

arXiv2018

Decomposing Heegaard splittings along separating incompressible surfaces in 3-manifolds

In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating surface. Further if the Heegaard surface is of Hempel distance at least 4, then there is a pair of such subsurfaces on both sides of the given separating surface. This gives a particularly simple hierarchy for the 3-manifold.

math.GT↗

arXiv2017

Z2-Thurston Norm and Complexity of 3-Manifolds, II

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 $\widetilde{\text{SL}_2(\mathbb{R})}.$

math.GT↗

arXiv2017

Multisections of piecewise linear manifolds

Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear $n$-manifold has a multisection, i.e. can be divided into $k+1$ $n$-dimensional $1$-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions and generalisations of our approach are given.

math.GT↗

arXiv2017

Computing trisections of 4-manifolds

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a $4$-manifold in terms of the number of pentachora ($4$-simplices) in a triangulation.

math.GT↗

arXiv2016

Minimal curvature-constrained networks

This paper introduces an exact algorithm for the construction of a shortest curvature-constrained network interconnecting a given set of directed points in the plane and an iterative method for doing so in 3D space. Such a network will be referred to as a minimum Dubins network, since its edges are Dubins paths (or slight variants thereof). The problem of constructing a minimum Dubins network appears in the context of underground mining optimisation, where the aim is to construct a least-cost network of tunnels navigable by trucks with a minimum turning radius. The Dubins network problem is similar to the Steiner tree problem, except that the terminals are directed and there is a curvature constraint. We propose the minimum curvature-constrained Steiner point algorithm for determining the optimal location of the Steiner point in a 3-terminal network. We show that when two terminals are fixed and the third varied, the Steiner point traces out a limaçon.

math.MG↗

arXiv2015

Thin position for incompressible surfaces in 3-manifolds

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to Heegaard splittings. For simplicity, we focus on the case of separating incompressible surfaces, since non-separating ones have been extensively studied. After putting the surfaces into Morse position relative to the height function associated to the Heegaard splittings, a thin position method is applied so that levels are thin or thick, depending on the side of the surface. The complete description of the surface in terms of these thin/thick levels gives a hierarchy. Also this thin/thick description can be related to properties of the curve complex for the Heegaard surface.

math.GT↗

arXiv2015

The classification of homotopy classes of bounded curvature paths

A bounded curvature path is a continuously differentiable piecewise $C^2$ path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our previous results here we finish a program started by Lester Dubins in 1961.

math.MG↗