New topological biquandles created using skew braces.
problem Creating nontrivial topological biquandles.
method Using the concept of skew braces.
result Constructs nontrivial examples of topological biquandles.
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48 results for “topological construction”
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
We review the AKSZ construction as applied to the topological open membranes and Poisson sigma models. We describe a generalization to open topological p-branes and Nambu-Poisson sigma models.
Novel M-theory approach classifies topological phases of matter.
problem Classifying and understanding topological phases of matter.
method Establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds, identifying topological phases from internal wrapped 3-manifolds.
result Paves a new route toward the classification of topological phases of matter, including fermionic and non-unitary phases.
Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…
Find limiting sets for digital cones and suspensions.
problem Digital topology cone and suspension constructions.
method Identify (m, n)-limiting sets, especially (0, 0)-freezing sets.
result Discover (0, 0)-limiting sets for digital cones and suspensions.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…
Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.
We construct infinitely many Legendrian links in the standard contact with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in that bound topologically distinct pieces of algebraic curves in , is applied to find contact 3-…
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
Isoperimetry by stretchingmath.DG
Constructs isoperimetric regions from separating hypersurfaces.
problem Finding isoperimetric regions in closed manifolds.
method Constructs isoperimetric regions from separating hypersurfaces.
result Yields isoperimetric boundaries with diverse topological types and singular sets.
Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTscond-mat.str-el
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the anomaly indicator for time-reversal symmetric topological superconductors.
Authors construct symplectic Lefschetz pencils on complex projective plane.
problem Construct symplectic Lefschetz pencils on complex projective plane.
method Differential topological construction, analogous to holomorphic pencils.
result Explicit monodromy factorization and topological construction for .
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
New framework for quantum invariants of 3-manifolds using homology.
problem Quantum invariants of 3-manifolds.
method Homological construction of mapping class group representations extended to cobordisms.
result Construction of a -dimensional TQFT assigning twisted homology to surfaces.
Constructs flows on manifolds with small curvature, proving Euclidean topology.
problem Geometric structure of manifolds with unbounded curvature.
method Distance like functions with integral hessian bound, Ricci flows.
result Manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors -regular and small curvature concentration are topologically Euclidean.
We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda--Kazez--Matic. This topological field theory stores inf…
Constructs dg categories from surfaces using Khovanov homology.
problem Categorify quantum topology using surfaces and Khovanov homology.
method Constructs dg categories from surfaces using structures in Khovanov homology.
result Unified perspective on various categorified quantum topology constructions.
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic …
Constructs a topological cover of real line's multiplicative group.
problem Topological cover of real line's multiplicative group.
method Homological algebra, 2D Lorentz geometry, high-school trigonometry.
result Interesting topological cover constructed.
Researchers provide a simple topological method for Burau representations of loop braid groups.
problem Constructing Burau representations of loop braid groups.
method Simple topological construction of the Burau representations.
result One of the representations is more subtle and topologically natural but not easily combinatorially obvious.
Constructing VAE Latent Spaces with Prescribed Topology
problem Resolving topological mismatch in VAEs for non-Euclidean data
method A constructive framework for product covering spaces
result Topology-aware latent representations with closed-form KL divergences
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with -twists and higher cohomology.
Topological Mirror Symmetrymath.AG
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…
Constructs new topological theories in 2D not fitting standard axioms.
problem Developing new topological theories in 2D that don't conform to traditional axioms.
method Universal construction by Blanchet et al., Kronecker's characterization, field extension, Hankel matrices, Schur polynomials, and foam evaluation.
result Introduction of non-multiplicative theories and classification over finite-dimensional state spaces.
To every oriented link , we associate a topologically defined biquandle , which we call the topological biquandle of . The construction of is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the top…
We construct a simple finite-dimensional topological quantum field theory for compact 3-manifolds with triangulated boundary.
On topological M-theoryhep-th
We construct a gauge fixed action for topological membranes on -manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on the theory naturally reduces…
Constructs Kleinian groups from free groups via hyperbolization.
problem Creating Kleinian groups from free groups.
method Direct product of rank 2 free groups and strict hyperbolization.
result Description of limit set and its topological dimension.
Topological model created for HOMFLY-PT polynomial from link diagrams.
problem Constructing categorifications for HOMFLY-PT polynomial.
method Explicit Lagrangian submanifolds on Heegaard surfaces.
result Invariants derived from link diagrams are given by intersections of submanifolds.
Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
Paper presents voxel graph operators for vector data models.
problem Efficient conversion and analysis of geometric models.
method Topological voxelization, graph construction, differential operator derivation.
result Discrete differential and integral operators from voxel complexes.
New tools for constructing fixed point sets in digital topology.
problem Constructing fixed point sets in digital topology.
method Defining excludable points and articulation points, and showing their exclusion from freezing sets.
result Excludable points and articulation points can be excluded from all freezing sets.
We show that infinitely many of the simply connected 4-manifolds constructed by Levine and Lidman that do not admit PL spines actually admit topological spines.
We show how topology of a space may lead to tensor fields on (the smooth part of) moduli spaces of the fundamental group.
New topological complexity measures for neural networks.
problem Measuring complexity of neural network functions.
method Generalized piecewise-linear Morse theory applied to ReLU networks.
result Local complexity can be arbitrarily high.
Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved th…
This paper introduces a new method for neural networks that doesn't need a global coordinate system.
problem The lack of a global coordinate system in neural networks limits their performance and explainability.
method Proposes a learnable topological layer that works in a general metric space (Hilbert space) without requiring a Euclidean space.
result The proposed method eliminates the need for a costly parametrization stage and achieves optimal network performance.
AI models help solve a symplectic topology problem.
problem Lagrangian smoothability question from Abouzaid et al.
method Used large language models (LLM) to approach the problem.
result Suggested new constructions in polyhedral symplectic topology.
New stable homotopy refinement of quantum annular Khovanov homology.
problem Quantum topological Hochschild homology and annular Khovanov spectra.
method Introducing quantum topological Hochschild homology (qTHH) and constructing a new stable homotopy refinement of quantum annular Khovanov homology.
result The new stable homotopy refinement agrees with qTHH of spectral Chen-Khovanov tangle bimodules and recovers earlier work.
A free Z-action by isometries on a compact metric space which is not embeddable into a cubical shiftmath.DS
New example solves topological dynamics problem.
problem Embedding compact metric space into cubical shift.
method Borsuk-Ulam theorem, p-adic completions, equivariant Sullivan conjecture.
result Existence of a compact metric space not embeddable into a cubical shift.
We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.
This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…
Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we constr…