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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 $11d$., Comment: Appearing at NeurIPS2022

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 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.

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., Comment: 61 pages, 7 figures, and 9 tables; V2 incorporates referee's comments; V3 numbering adjusted to match published version; to appear in Inventiones Mathematicae

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 $\mathcal{T}_{\Lambda}$, there is a bijective correspondence between the closed normal surfaces in $\mathcal{T}^*$ and the closed normal surfaces in $\mathcal{T}_{\Lambda}$ 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 $\mathcal{T}_{\Lambda}$ 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., Comment: 21 pages, 6 figures; revised and improved version of an earlier paper arXiv:1108.2936, Annular efficient triangulations of 3-manifolds

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.

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., Comment: Minor corrections. To appear in L'Enseignement Math\'ematique. 41 pages, 42 figures

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., Comment: 6 pages, 2 figures

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

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., Comment: 15 pages, 9 figures

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 lima\c{c}on., Comment: 20 pages, 16 figures

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., Comment: 23 pages, 6 figures

The installation of truck turntables at above-ground crushers has the potential to yield gains in efficiency and productivity by eliminating the need for the trucks to turn around and reverse up to the crusher to dump their loads. The benefits include savings in time, fuel consumption and tyre wear. In addition, having a smaller area near the crushers dedicated to manoeuvring the trucks can be a significant benefit at some sites. Safety is also enhanced using turntables. This paper describes a project with the Australian Turntable Company (ATC) to develop software that generates near-optimal layouts for the paths taken by the trucks in the vicinity of the crushers when turntables are present or absent. The geometry of the paths is constrained by the turning radius of the trucks and by the boundary of the manoeuvring area. The software quantifies the benefit of installing turntables by generating a design for paths with turntables and comparing it with the layout in current or planned use by the mining company. Through the use of this software tool, a mining company can readily assess the benefits of using turntables at their site.

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., Comment: 14 pages, 2 figures

Many Ore Reserves are harvested by a combination of open pit and underground mining methods. In these cases there is often material that could be mined by either method, and a choice has to be made. The area containing this material is referred to as the transition zone. Deciding where to finish the open pit and start the underground is referred to as the transition problem and it has received some attention in the literature since the 1980s. In this paper we provide a review of existing approaches to the transition problem encompassing: graph-theory based optimisation employing an opportunity cost approach; heuristics and integer programming. We also present a novel opportunity cost approach, allowing it to take into account a crown pillar, and show how the new approach can be best applied through the unconventional application of an existing mine optimisation tool.

We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3-manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3-manifold is essential or strongly essential are given., Comment: 30 pages, 14 figures. Exposition improved

This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken n-manifolds, an interesting connection between very short hierarchies and even triangulations is observed., Comment: 28 pages, 6 figures

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., Comment: To appear in the Israel Journal of Mathematics. arXiv admin note: substantial text overlap with arXiv:1403.4930, arXiv:1403.4911

Consider two elements in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{\mathbb R}^2$. In this work we address the problem of characterizing the paths of bounded curvature and minimal length starting at $x$, finishing at $y$ and having tangents at these points $X$ and $Y$ respectively. This problem was first investigated in the late 50's by Lester Dubins. In this note we present a constructive proof of Dubins' result giving special emphasis on the geometric nature of this problem., Comment: 14 pages, 8 figures

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k., Comment: 22 pages, 2 figures

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index structure if and only if T is 1-efficient and (b) if M is hyperbolic, it has a canonical set of 1-efficient ideal triangulations related by 2-3 and 0-2 moves which preserve the 3D index. We illustrate our results with several examples., Comment: 60 pages, 27 figures

Starting with an ideal triangulation of the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulations of M itself. Besides a step-by-step algorithm for such a construction, we provide examples of an inflation of the two-tetrahedra ideal triangulation of the complement of the figure-eight knot in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the figure-eight knot exterior. As another example, we provide an inflation of the one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary. Several applications of inflations are discussed., Comment: 48 pages, 45 figures

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every knot or link complement in the 3-sphere has such a triangulation. We also give an example of a triangulation without a strict angle structure, where the obstruction is related to the homology hypothesis, and an example illustrating that the triangulations produced using our methods are not generally geometric., Comment: 28 pages, 9 figures. Minor edits and clarification based on referee's comments. Corrected proof of Lemma 7.4. To appear in the Journal of Topology

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle. Our main results are 1. The Smale Conjecture holds for all elliptic 3-manifolds containing geometrically incompressible Klein bottles. These include all quaternionic and prism manifolds. 2. The Smale Conjecture holds for all lens spaces L(m,q) with m at least 3. These results complete the Smale Conjecture for all cases except the 3-dimensional real projective space and those admitting a Seifert fibering over the 2-sphere with three exceptional fibers of types (2,3,3), (2,3,4), or (2,3,5). The technical work needed for these results includes the result that if V is a Haken Seifert-fibered 3-manifold, then apart from a small list of known exceptions, the inclusion from the space of fiber-preserving diffeomorphisms of V to the full diffeomorphism group is a homotopy equivalence. This has as a consequence: 3. The space of Seifert fiberings of V has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background material on diffeomorphism groups is included., Comment: 177 pages, 23 figures

A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, irreducible, boundary-irreducible, and an-annular 3-manifold, any triangulation can be modified to an annular-efficient triangulation. It follows that for a manifold satisfying this hypothesis, there are only a finite number of boundary slopes for incompressible and boundary-incompressible surfaces of a bounded Euler characteristic., Comment: 21 pages, 6 figures

Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this question and prove it for all knots up to 11 crossings except for two examples. We also establish the conjecture for all Montesinos knots and for all generalized arborescently alternating knots. For knot exteriors containing closed incompressible surfaces satisfying a simple homological condition, we establish that the knots satisfy the Neuwirth conjecture. If there is a proper degree one map from knot $K$ to knot $K'$ and $K'$ satisfies the Neuwirth conjecture then we prove the same is true for knot $K$. Algorithms are given to decide if a knot satisfies the various versions of the Neuwirth conjecture and also the related conjectures about whether all non-trivial knots have essential surfaces at integer boundary slopes., Comment: 31 pages, 22 figures

Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, giving an example of a veering taut triangulation that is not layered., Comment: 15 pages, 9 figures

The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of hyperplanes in Hyperbolic space, and Piecewise-Linear hyperplane arrangements, are shown to represent maximum classes, generalizing the corresponding Euclidean result. A main result is that PL arrangements can be swept by a moving hyperplane to unlabeled d-compress any finite maximum class, forming a peeling scheme as conjectured by Kuzmin & Warmuth. A corollary is that some d-maximal classes cannot be embedded into any maximum class of VC dimension d+k, for any constant k. The construction of the PL sweeping involves Pachner moves on the one-inclusion graph, corresponding to moves of a hyperplane across the intersection of d other hyperplanes. This extends the well known Pachner moves for triangulations to cubical complexes., Comment: 37 pages, 18 figures, submitted to the Journal of Machine Learning Research

In this paper we settle Thurston's old question of whether the Weber-Seifert dodecahedral space is non-Haken, a problem that has been a benchmark for progress in computational 3-manifold topology over recent decades. We resolve this question by combining recent significant advances in normal surface enumeration, new heuristic pruning techniques, and a new theoretical test that extends the seminal work of Jaco and Oertel., Comment: 22 pages, 10 figures, 3 tables; v2: expanded introduction; v3: minor revisions (accepted for Trans. Amer. Math. Soc.)

A new lower bound on the complexity of a 3-manifold is given using the Z2-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs., Comment: 19 pages, 4 figures

This is a personal view of some problems on minimal surfaces, Ricci flow, polyhedral geometric structures, Haken 4-manifolds, contact structures and Heegaard splittings, singular incompressible surfaces after the Hamilton-Perelman revolution., Comment: This is the version published by Geometry & Topology Monographs on 3 December 2007

This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L(4k, 2k-1)$ and the generalised quaternionic space $S^3/Q_{4k}$ have complexity $k,$ where $k\ge 2.$ Moreover, it is shown that their minimal triangulations are unique., Comment: 8 pages, 3 figures

The linear potential binding a quark and an antiquark in mesons is generalized to baryons and multiquark configurations as the minimal length of flux tubes neutralizing the color, in units of the string tension. For tetraquark systems, i.e., two quarks and two antiquarks, this involves the two possible quark--antiquark pairings, and the Steiner tree linking the quarks to the antiquarks. A novel inequality for this potential demonstrates rigorously that within this model the tetraquark is stable in the limit of large quark-to-antiquark mass ratio., Comment: 8 pages, 6 figures, pdflatex

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1, the minimal layered triangulation of the lens space L(2n,1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens spaces L(2n,1)., Comment: 32 pages, 6 figures

We examine the behavior of a one-dimensional superconducting wire exposed to an applied electric current. We use the time-dependent Ginzburg-Landau model to describe the system and retain temperature and applied current as parameters. Through a combination of spectral analysis, asymptotics and canonical numerical computation, we divide this two-dimensional parameter space into a number of regions. In some of them only the normal state or a stationary state or an oscillatory state are stable, while in some of them two states are stable. One of the most interesting features of the analysis is the evident collision of real eigenvalues of the associated PT-symmetric linearization, leading as it does to the emergence of complex elements of the spectrum. In particular this provides an explanation to the emergence of a stable oscillatory state. We show that part of the bifurcation diagram and many of the emerging patterns are directly controlled by this spectrum, while other patterns arise due to nonlinear interaction of the leading eigenfunctions.

This paper has been withdrawn by the author, as the proof of Theorem 3.2 contains a flaw; subsequently, both it and Theorem 3.3 are not known to hold. The content of Section 5 has been improved and expanded upon in two separate papers. The structural properties in 5.1 and 5.2 appear in arXiv:1101.2603; the classification in 5.3 in arxiv:1101.2890. Please refer to Proposition 4.4 in the former as a reference for arXiv:0807.4795v1., Comment: This paper has been withdrawn

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-disk; if a meridian slope is given, then it can be determined if a longitude bounds an embedded punctured-disk. The methods use normal surface theory but do not follow the classical approach. Properties of minimal vertex triangulations, layered-triangulations, 0--efficient triangulations and especially triangulated Dehn fillings are central to our methods. We also use an average length estimate for boundary curves of embedded normal surfaces; a version of the average length estimate with boundary conditions also is derived. An algorithm is given to construct precisely those slopes on the boundary of a given link-manifold that bound an embedded punctured-disk in the link-manifold., Comment: 44 pages, 9 figures

Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover \pi:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g(S) - 3. We also get a bound on the complexity of the branch set in terms of g(S). If we assume that M is non-Haken, by Casson and Gordon we may replace g(S) by g(M) in all the statements above., Comment: Minor changes only. Final version, to appear in Communications in Analysis and Geometry