GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
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48 results for “handle theory”
Handle homology of manifoldsmath.GT
We give an entirely geometric proof, without recourse to cellular homology, of the fact that in the chain complex defined by a handle decomposition of a given manifold. Topological invariance of the resulting `handle homology' is a consequence of Cerf theory.
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problemsmath.GT
Review of gem theory's interactions with Kirby diagrams and trisections.
problem Representing and understanding PL 4-manifolds.
method Combining gem theory with Kirby diagrams and trisections.
result New gems representing 4-manifolds requiring 3-handles.
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
Paper introduces tensor product of quandles for knot classification.
problem Classifying knot invariants of surface-links with 1-handles.
method Introduces tensor product of quandles and applies to surface-links.
result Tensor product of knot quandles/classical quandles can classify surface-link invariants.
In this paper, we introduce a deformation analysis of index theory over non compact manifolds, by use of new functional spaces which are the reduced version of Sobolev spaces. It allows to construct Fredholm theory for elliptic differential operators over non compact spaces which possibly do not have closed range with …
Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.
problem Understanding handle numbers and incompressible Seifert surfaces in sutured manifolds and nearly fibered knots.
method Extending Haken's Theorem, analyzing product annuli and disks, and examining specific knot types.
result Variety of handle numbers and unique incompressible Seifert surfaces in nearly fibered knots.
Classical Knot Theorymath.GT
This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.
Complexity of 3-orbifoldsmath.GT
We extend Matveev's theory of complexity for 3-manifolds, based on simple spines, to (closed, orientable, locally orientable) 3-orbifolds. We prove naturality and finiteness for irreducible 3-orbifolds, and, with certain restrictions and subtleties, additivity under orbifold connected sum. We also develop the theory of…
After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, an…
The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
Framework extends neural operators to handle functions outside training set.
problem Robust handling of functions beyond the training set.
method Kernel approximation techniques and Reproducing Kernel Hilbert Spaces (RKHSs) theory.
result Theoretical framework and empirical validation for reliable function extension.
The paper studies Morse theory for Lie algebra actions on Riemannian foliations.
problem Morse theory for Lie algebra actions on Riemannian foliations.
method Equivariant Morse-Bott theory on leaf space.
result Established foliated versions of Morse-Bott lemma and handle presentation theorem.
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.
problem Representing compact 4-manifolds using Kirby diagrams and graphs.
method Algorithmically constructing 5-colored graphs from Kirby diagrams to represent PL 4-manifolds.
result Upper bounds for gem-complexity and regular genus derived from Kirby diagrams.
New findings about twists in 4-sphere diffeomorphisms.
problem Understanding isotopy classes of diffeomorphisms in 4-sphere.
method Using Cerf theory and twists along Montesinos twins.
result The subgroup of twists along Montesinos twins is trivial or cyclic of order two.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
Optimizes portfolios using CPT utility via convex optimization.
problem Maximizing CPT utility in portfolio selection.
method Minorization-maximization (MM) algorithm and convex-concave (CC) procedure.
result Problems can be solved globally and efficiently.
Study Morse complexity of manifolds and homology classes, proving bounds and implications.
problem Understanding Morse complexity of manifolds and homology classes.
method Used surgery theory and index theory to prove upper and lower bounds.
result Locally symmetric spaces of Lie groups with discrete series representations do not admit open book decompositions.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
problem Mapping discrete Morse functions to shellings for topological analysis.
method Inducing Morse shellings on the second barycentric subdivision of a simplicial complex.
result Critical tiles of induced shellings correspond to critical faces of the discrete Morse function.
Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.
Surveying risk measures for handling uncertainty in various fields.
problem Handling uncertainty in engineering and data-driven problems.
method Review of risk measures and their applications.
result Rapid development and widespread use of risk measures.
Kernel ridge regression imputation with consistent variance estimation for handling missing data.
problem Handling missing data in statistical analysis.
method Kernel ridge regression imputation combined with entropy method for variance estimation.
result Root-n consistency of the imputation estimator in a Sobolev space setting.
New invariant structure found near a knot in flows.
problem Finding additional invariant structure in flows with known knots.
method Developed a 'coloured' handle theory to prove the existence of additional invariant structure.
result Additional invariant structure found arbitrarily close to a known invariant knot.
We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits…
Aspherical -pairsmath.GT
The study of -pairs extends results for aspherical 3-manifolds.
problem Understanding -pairs with aspherical ambient spaces.
method Attaching 1-handles to -pairs with aspherical ambient space and -injective boundary.
result There are only finitely many -pairs with a specific group property.
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index . Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental gro…
The paper develops methods to handle missing data using regularized M-estimation in reproducing kernel Hilbert space.
problem Handling missing data in statistical analysis.
method Kernel ridge regression for imputation and maximum entropy method for propensity score estimation.
result The proposed methods achieve statistical consistency and asymptotic equivalence.
Study handles in 4-manifolds with cyclic fundamental group.
problem Understanding handle decompositions for specific 4-manifolds.
method Constructed handle decompositions based on fundamental group and Betti numbers.
result Handle decomposition formula for specific 4-manifolds.
We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete …
Elliptic surfaces without 1-handles proven for specific cases.
problem Proving elliptic surfaces have no 1-handles.
method Analyzing handle decompositions of elliptic surfaces for specific .
result Proven existence of handle decompositions without 1-handles for specific elliptic surfaces.
Theory for deep neural network approximation of score function and its derivatives.
problem Handling data distributions with low-dimensional structure and unbounded support.
method Simultaneous approximation of the score function and its derivatives using deep neural networks.
result Approximation error bounds match literature but relax bounded support requirement.
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing -component links with a Dehn surgery realizing . We call such links R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diff…
Proves finiteness and holonomicity of skein modules for 3-manifolds.
problem Finiteness and holonomicity of skein modules for 3-manifolds.
method Defining skein transfer bimodules and using q-analogues of D-module theory.
result Internal skein modules are holonomic modules over the internal skein algebra of the boundary.
Study of 3-handle attachments in 4D manifolds using Kirby calculus.
problem Inclusion of 3-handles in Kirby diagrams for 4D manifolds.
method Developed moves to extend Kirby calculus, established homological criterion.
result Identified geometric basis of 3-handle attachments, proved uniqueness theorem.
New method connects compression bodies through cone manifolds.
problem Understanding how compression bodies can be transformed.
method Using cone manifold holonomy groups and standard space techniques.
result Realized all edges in compression body graph through paths.
The study shows conditions for elliptic surfaces without 1-handles.
problem Finding conditions for elliptic surfaces without 1-handles.
method Analyzing knot surgeries and log-transformations of elliptic surfaces.
result Conditions for elliptic surfaces without 1-handles are established.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that …
We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss can control the test error under all Moreau envelopes of the loss . We use our finite-sample bound to directly recover…
The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained --- they correspond to open handlebodies with all handles of index lessthan or = 2. An uncountable collection of exotic R^4's is shown to ad…
New algorithms handle phase retrieval with rank d measurements, revealing phase transitions.
problem Phase retrieval with rank d measurements.
method Random duality theory (RDT) and descending phase retrieval algorithms (dPR).
result Minimal sample complexity ratio for dPR's success exhibits phase transitions.
We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classic…
The paper tackles noisy labels in high-dimensional data, showing low-dimensional intuitions fail and proposing an optimized method.
problem Noisy labels in high-dimensional data classification.
method Linear classifier with a label noisiness aware loss function, using random matrix theory and Gaussian mixture data model.
result The performance of the linear classifier in high-dimension converges to a limit involving scalar statistics of the data, and the optimal classifier in low-dimension fails.
We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and with relations well known from the 3-dimensional theory of finite type invariants…