Four counterexamples in surface homology.
problem Understanding surface homology and minimal homology bases.
method Presenting four specific counterexamples.
result Minimal homology bases can be challenging to work with.
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48 results for “homology bases”
Remarks on some infinitesimal symmetries of Khovanov--Rozansky homologies in finite characteristicmath.GT
New proof of Khovanov-Rozansky homology base point independence in finite characteristic.
problem Base point independence of Khovanov-Rozansky homology in characteristic .
method New proof using infinitesimal symmetries of -homology.
result Proof of base point independence in characteristic .
The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the…
Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
Persistent homology can recognize knotting in curves.
problem Recognizing knotting in curves
method Compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score.
result Systematic differences between knotted and unknotted structures are revealed.
We present three equivalent definitions of -equivariant symplectic homology. We show that, using rational coefficients, the positive part of -equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduc…
The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.
problem Calculating Witten-Reshetikhin-Turaev invariants for Seifert fibered homology 3-spheres.
method Explicit modular transformation formulas of homological blocks.
result New proof of Witten asymptotic conjecture for Seifert fibered homology 3-spheres.
We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show…
Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known that it is equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurst…
Classifies links with small Khovanov homology ranks.
problem Classifying links with specific ranks in Khovanov homology.
method Previous results combined with new classifications.
result All links with ranks ≤ 8 and three-component links with ranks ≤ 12 are classified.
New spectral sequence connects Khovanov homology to real monopole Floer homology.
problem Computing real monopole Floer homology for links.
method Defined a spectral sequence from reduced Khovanov homology of the mirror link.
result Spectral sequence abuts to real monopole Floer homology of a link.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
Researchers describe a new method to compute Seiberg-Witten-Floer spectra for a specific class of manifolds.
problem Computing Seiberg-Witten-Floer spectra for a specific class of manifolds.
method Using lattice homology, they provide an explicit combinatorial description of the spectra.
result They calculate Manolescu's κ-invariant for certain connected sums of the spaces.
We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of 3-manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associa…
Heegaard Floer homology connects group presentations to algebraic invariants.
problem Understanding algebraic invariants of groups through geometric topology.
method Using Heegaard Floer homology, linking group presentations to algebraic invariants.
result Heegaard Floer homology groups are independent of choices and invariant under stable transformations.
Lectures on knot homologymath-ph
We provide various formulations of knot homology that are predicted by string dualities. In addition, we also explain the rich algebraic structure of knot homology which can be understood in terms of geometric representation theory in these formulations. These notes are based on lectures in the workshop "Physics and Ma…
We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a s…
Link homology compared with geometric link invariants using Bott-Samelson varieties.
problem Comparing different link homology theories with geometric link invariants.
method Using Khovanov-Rozansky homology and equivariant cohomology applied to Bott-Samelson varieties.
result Equivariant integral sl(n) link homology with specialized or universal potential.
We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D^{2} \times I) in terms o…
Proves a formula for instanton Floer homology of knots.
problem Calculating changes in instanton Floer homology for knots.
method Uses sutured instanton homology and the octahedral lemma.
result Derives an integral surgery formula for framed instanton homology.
For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. T…
Lecture notes on link homologies and knotted surfaces, focusing on 4D obstructions.
problem Understanding knotted surfaces using link homology theories.
method Overview of link homology theories, introduction to Khovanov homology, and computational techniques.
result New insights into the homomorphisms assigned to link cobordisms for knotted surfaces.
We prove that every H-thin link has no -torsion for in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is bas…
We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…
In the present paper, we construct the Khovanov homology theory for virtual links. Besides the direct approach with Z_{2} coefficients we also describe the Khovanov homology for framed links and the Khovanov homology using ``double cover''. The latter two approaches are based on the notion of ``atom''.
Study initiates homology theory for Bol-Moufang quasigroups.
problem Developing a homology theory for a specific type of quasigroups.
method Using extensions by affine quasigroups, define boundary operations and compute homology groups.
result Computed second homology groups for examples, speculating on relation to functorial homology.
Braid Floer homologymath.DS
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…
New homology theories for orbifolds and weighted polyhedra.
problem Homology of orbifolds and weighted polyhedra.
method Introducing AW-homology and DW-homology from special triangulations.
result Invariant under orbifold isomorphisms and generalized Poincaré duality.
New invariants derived from link homology for 4-manifolds.
problem Creating invariants for surfaces in smooth 4-manifolds.
method Skein lasagna modules based on equivariant and deformed link homology.
result Non-vanishing and decomposition results for deformed skein lasagna modules.
Paper reviews Viro's definition of Khovanov homology for tangles.
problem Defining Khovanov homology for unoriented framed tangles.
method Explains Viro's definition and the relation between -matrix and Kauffman bracket.
result Clarifies the relation between -matrix and Kauffman bracket in Khovanov homology.
New metric improves clustering in persistent homology.
problem Improving clustering accuracy in persistent homology.
method Defined a new non-archimedean cophenetic metric.
result Cophenetic metric enhances clustering quality and inter-relations.
The paper identifies knots in specific lens spaces based on their complements.
problem Determining knots in homology lens spaces based on their complements.
method Analyzing knots in homology lens spaces with specific conditions.
result Knots are determined by their complements under certain conditions.
Recover manifold homology from a sample in Euclidean space.
problem Recovering the homology of an immersed manifold from a sample.
method Estimate tangent bundle, apply measure-based filtrations for persistent homology, use normal reach.
result Construction consistent and stable, does not require manifold dimension.
Proves a conjecture for a specific group using spectral sequences and homology.
problem Proves the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4.
method Used the Adams spectral sequence and detection theorems to compute connective real k-homology.
result Determines differentials of the Adams spectral sequence and studies the cap structure of relevant sub-hopf algebras.
New method connects curvature and Persistent Homology for networks.
problem Efficient computation of Persistent Homology for complex networks.
method Discrete Morse Theory, Bloch's extension, Forman-Ricci curvature.
result Efficient Persistent Homology scheme using curvature-based approach.
It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzb…
Khovanov homology is a recently introduced invariant of oriented links in . It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of the Khovanov homology. Based on ou…
Refines Khovanov homology using signed Burnside categories.
problem Stable homotopy refinement of Khovanov homology.
method Signed Burnside category approach to compare Blanchet and Khovanov chain complexes.
result Stable homotopy type construction for link diagrams.
Differentiable topology layer for machine learning models.
problem Regularizing and incorporating topological priors in machine learning models.
method Computes persistent homology based on level set and edge-based filtrations.
result Demonstrates three novel applications of the topology layer in machine learning.
New clustering algorithm uses persistent homology for space-time data.
problem Clustering space-time data without labeled examples.
method Persistent homology for topological data analysis, analyzing data at multiple resolutions.
result The algorithm distinguishes true features from noise based on persistence.
Clarifies structures in link homology theories using Frobenius extensions.
problem Understanding key flavors of equivariant SL(2) link homology theories.
method Provides a convenient scheme and diagrammatics for Frobenius extensions.
result Proposes a setup for working over non-degenerate base rings.
In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology.…
We compute persistent homology using an intrinsic metric derived from density.
problem Estimating topological features from high-dimensional data.
method Density-based metric learning for persistent homology.
result Persistent homology converges to intrinsic manifold metric.
Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.
problem Lipschitz homology classes and their moduli.
method Defining a new modulus and proving a homological duality theorem.
result Every relative Lipschitz -homology class has a unique dual class satisfying a specific modulus product equality.
The paper constructs semistrict monoidal 2-categories from foam evaluations.
problem Creating examples of semistrict monoidal 2-categories.
method Using a closed foam evaluation formula as input, the paper rigorously constructs semistrict monoidal 2-categories.
result The constructed monoidal 2-categories are semistrict, have duals and adjoints, and carry a spatial duality structure.
Survey of contact homology for contact manifolds.
problem Understanding contact manifolds through homology.
method Holomorphic curves invariants.
result Useful for students and young mathematicians.