Algorithm simplifies triangulations near curves efficiently.
problem Efficiently simplifying triangulations near given curves.
method Uses flips and powers of Dehn twists in polynomial time.
result Algorithm completes in polynomial time based on curve size.
Algorithm simplifies geometric intersection computation.
problem Computing geometric intersection between curves.
method Linear programming and triangulation simplification.
result Algorithm runs in polynomial time.
Every noncompact surface has a 3-rigid triangulation.
problem Classifying noncompact surfaces and proving their rigidity.
method Triangulation and minimally rigid structures.
result Every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid.
We show that every semialgebraic set admits a semialgebraic triangulation such that each closed simplex is C1 differentiable. As an application, we give a straightforward definition of the integration ∫Xω over a compact semialgebraic subset X of a differential form ω on an ambient algebraic manifold, that…
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
It is important to have fast and effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental …
It is important to have effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work that…
Constructs simplified or complexified simplicial complexes.
problem Efficiently simplifying or complexifying complex spaces.
method Embeddings of simplicial complexes into a simplicial ball with bounded degrees and low volume.
result Realizes complicated spaces as parts of a ball/sphere or gives spheres specific metrics.
New method simplifies 3-manifold Heegaard genus computation.
problem NP-hardness of computing Heegaard genus of 3-manifolds.
method Avoiding almost normal surfaces by modifying triangulations.
result Successfully computed Heegaard genus for 2,705 out of 3,000 manifolds.
Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognit…
ReAPR simplifies hard unknots by reembedding and rerouting, revealing hidden simplifications.
problem Training AI to recognize knots, especially hard unknots, is challenging.
method Alternates pass-move reduction with geometric re-embedding, minimizing total variation of a height function.
result ReAPR successfully simplifies hard unknots, including Kauffman's challenge unknots, in under 30 seconds.
Researchers compute A-polynomials of manifolds using symplectic properties and cluster algebras.
problem Computing A-polynomials of infinite families of knots and related manifolds is difficult.
method Starting with a triangulation, they use symplectic properties of the Neumann-Zagier matrix to simplify the computation.
result The defining equations of A-polynomials of manifolds obtained by Dehn filling are Ptolemy equations.
The paper defines useful model simplifications and calibration methods for predictive uncertainty.
problem Quantifying predictive uncertainty in models that simplify data and predictions.
method Defines conditions for useful model simplifications and calibration schemes.
result Conditions for when simplified models can be used and when they should be ignored.
We simplify singularities of Lagrangian and Legendrian fronts without homotopy obstructions.
problem Simplifying singularities of Lagrangian and Legendrian fronts without homotopy obstructions.
method Establishing a full h−principle for simplification of singularities of Lagrangian and Legendrian fronts. result Simplification of singularities can be achieved by means of a Hamiltonian isotopy if there are no homotopy theoretic obstructions.
This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diag…
We prove that any arc-presentation of the unknot admits a monotonic simplification by elementary moves; this yields a simple algorithm for recognizing the unknot. We obtain similar results for split links and composite links.
Natural gradient simplification for deep learning networks.
problem Efficiency in training deep Bayesian networks.
method Analysis of two geometries of Fisher information matrix and development of a method to simplify natural gradient for the second geometry.
result A method to simplify natural gradient for deep networks using an auxiliary recognition model.
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a mi…
Recent work on stable minimal hypersurface singularities.
problem Understanding singularities of stable minimal hypersurfaces.
method Simplifications of technical discussion in previous work.
result Simplified approach to analyzing hypersurface singularities.
This paper simplifies deep ReLU networks into local linear models for better interpretability.
problem Limited transparency and interpretability of deep neural networks, especially ReLU networks.
method Local linear representation and equivalent set of local linear models (LLMs).
result Simplified deep ReLU networks for better interpretability and diagnostics.
Simplifies residual flows to make flow-based modeling more practical.
problem Extremely high computational cost of residual flows limits their applicability.
method Introduces Quasi-Autoregressive (QuAR) approach to residual flows.
result Significantly reduces compute time and memory requirements for flow-based modeling.
The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplifications of link diagrams. In this pap…
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Efficient triangulations help in understanding 3-manifold boundaries.
problem Understanding boundary slopes in 3-manifolds.
method Introducing and studying boundary-efficient triangulations and inflating ideal triangulations.
result There are only finitely many boundary slopes for incompressible and \(\partial\)-incompressible surfaces in compact 3-manifolds.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…
Geometric triangulations can be transformed by bistellar moves.
problem Transforming geometric triangulations of different manifolds.
method Using bistellar moves, a type of local change to triangulations.
result Geometric triangulations of compact manifolds can be connected by bistellar moves.
In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible t…
AST provides a method to validate safe autonomy without unsafe simplifications.
problem Validation of safe autonomy in complex systems.
method Adaptive Stress Testing (AST) approach.
result AST can find failures without unsafe simplifications.
Triangulations without degree one edges are connected via moves.
problem Connectivity of triangulations without degree one edges.
method 2-3 and 3-2 moves.
result Subgraph of Pachner graph without degree one edges is connected.
NeuRewriter learns to choose and rewrite heuristics in combinatorial problems.
problem Time-consuming tuning of heuristics in combinatorial optimization.
method NeuRewriter uses reinforcement learning to learn a policy for picking heuristics and rewriting solutions.
result NeuRewriter outperforms existing methods in various combinatorial tasks.
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
Minimal triangulations for 229 hyperbolic census knots discovered.
problem Finding minimal triangulations for hyperbolic census knots.
method Ideal triangulations of the magic manifold, low-complexity triangulations for partial fillings, sorting into families.
result Minimal triangulations for 229 hyperbolic census knots discovered, conjectured to be minimal for all 42 families.
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, givin…
New method simplifies data analysis.
problem Complex data analysis challenges.
method Innovative algorithm for data simplification.
result Significant reduction in analysis time.
Smooth structures enable solving PDEs via optimized triangulations.
problem Solving partial differential equations (PDEs) on complex spaces.
method Introducing Frölicher space structure on CW complexes and spaces of triangulations to enable differential methods.
result Optimized triangulations can be used to solve standard PDEs.
Authors find small triangulations for specific 4-manifolds.
problem Finding optimal triangulations for 4-manifolds.
method Triangulated connected sums of CP^2 and S^2×S^2, conjectured minimal pentachora.
result Triangulations have the smallest number of pentachora for their types.
Paper tackles morphology simplification for Chinese-Spanish machine translation.
problem Challenges in morphology generation for unbalanced languages in machine translation.
method Proposes a new neural architecture for morphological simplification, combining embedding, convolutional, and recurrent neural network layers.
result Obtains over 98% accuracy in gender classification, over 93% in number classification, and an overall translation improvement of 0.7 METEOR.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
Minimal ideal triangulations studied for hyperbolic 3-manifolds.
problem Finding minimal triangulations of hyperbolic 3-manifolds.
method Characterization of low degree edges, layered solid torus subcomplexes, and 1-dimensional cohomology.
result Monodromy ideal triangulations of once-punctured torus bundles are minimal.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
Experimental results on veering triangulations of 3-manifolds.
problem Understanding the combinatorial structure of veering triangulations.
method Algorithmic construction and experimental analysis.
result Experimental insights into the structure of veering triangulations and their relation to topological invariants.
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surfac…
Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.
problem Finding the minimum number of tetrahedra in triangulations of 3-manifolds.
method Computed the triangulation complexity of all elliptic and sol 3-manifolds, within a bounded error.
result Computed the triangulation complexity of all elliptic and sol 3-manifolds.