Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
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48 results for “homology computation”
Method for computing Khovanov homology of tangles.
problem Limited explicit computational studies of Khovanov homology for tangles.
method Arc reduction approach to compute Khovanov homology.
result Derived and computed Poincaré polynomials for simple and complex tangles.
Computations for prime knots up to 11 crossings.
problem Computing HOMFLY homology for prime knots.
method Direct computations for all prime knots up to 11 crossings.
result HOMFLY homology determined for all prime knots up to 11 crossings.
Regularizes persistent homology gradients for neural network integration.
problem Ill-posed inverse problem in computing gradients of persistent homology.
method Regularization through a grouping term to define gradients for larger entities.
result Ensures gradients are defined with respect to larger entities, not individual points.
The paper simplifies the computation of a complex polynomial using Yang-Baxter operators.
problem Computing the homology of Yang-Baxter operators for arbitrary m.
method Reduced the computation to initial conditions and produced explicit formulas.
result Explicit formulas for the third and fourth homology.
Survey of computations for Riemann surface moduli spaces, focusing on unstable homology.
problem Computing unstable homology of moduli spaces of Riemann surfaces.
method Integral, mod-2, and rational coefficient computations; use of homology operations.
result Explicit generators of unstable homology for most cases determined.
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
Computes a specific homology for a type of braid.
problem Calculating a homology for a particular class of braids.
method Uses Khovanov-Rozansky homology for Coxeter braids on 4 strands.
result Computed the triply graded Khovanov-Rozansky homology.
Investigates polynomial time algorithms for computing Khovanov homology of braids.
problem Computing Khovanov homology for general braids is intractable.
method Examines polynomial time algorithms for 3-braids and a variation of the scanning algorithm for more general braids.
result Shows that for 3-braids, Khovanov homology can be computed in polynomial time, while for more general braids, it can be computed in polynomial time for bounded homological degrees.
New method uses algebras to speed up link Floer homology calculations.
problem Computing link Floer homology efficiently.
method Using bordered algebras to compute link Floer homology.
result Fast computation of the Thuston polytope for links.
Authors compute stable homology of torus knots using a new deformation technique.
problem Computing stable homology of torus knots.
method Link-splitting deformation (y-ification) of link homology.
result Explicit computation of -ified stable Khovanov--Rozansky homology of torus knots.
Computes quandle associated groups using group homology.
problem Computing associated groups of quandles.
method Using group homology theory to describe and compute associated groups.
result Computed second quandle homology groups of specific quandle families.
We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However, that approach was unable to explicitly identify the invariant of transverse links …
We compute Khovanov homology for tangles using TQFT.
problem Khovanov homology for tangles is not well studied or computed.
method Topological Quantum Field Theory (TQFT) construction.
result A comprehensive method for computing Khovanov homology of tangles.
New method computes first Vassiliev derivative of Khovanov homology.
problem Computing Vassiliev derivatives of Khovanov homology.
method Developed a crux complex to compute the first derivative.
result Direct computation of the first derivative of Khovanov homology.
In this paper, we give an algorithm to compute the hat version of the Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtrations coming from a null-homologous link in a three-manifold.
Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.
problem Computing Heegaard Floer homologies of Brieskorn spheres.
method Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong.
result Brieskorn spheres generate infinite rank summands in the homology cobordism group.
The pong algebramath.GT
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
Computes homology of an obstruction chain complex in grid homology.
problem Computing the homology of an obstruction chain complex in grid homology.
method Defined and computed the homology of the obstruction chain complex of the full grid.
result Results about the existence of sign assignments in grid homology.
Survey of bordered Floer homology for 3-manifolds with boundary.
problem Computing invariants for 3-manifolds with boundary.
method Extends Heegaard Floer homology to include boundary.
result Powerful computational tools for 3-manifolds with boundary.
Computes upper bounds for instanton knot homology.
problem Computing the exact dimension of instanton knot homology.
method Developed an algorithm using knot diagrams and Heegaard diagrams.
result Algorithm provides sharp bounds for knots up to seven crossings.
Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
We extend an approach of Beliakova for computing knot Floer homology and implement it in a publicly available computer program. We review the main programming and optimization methods used. Our program is then used to check that the Floer homology of a prime non-alternating knot with less than 12 crossings has no torsi…
We introduce a local algorithm for Khovanov Homology computations - that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational effici…
Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khova…
In this paper, we study the Khovanov homology of cable links. We first estimate the maximal homological degree term of the Khovanov homology of the (, )-torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the (, )-torus li…
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.
Adapts scanning algorithm for odd Khovanov homology.
problem Computing odd Khovanov homology efficiently.
method Uses mapping cone construction instead of tensor product.
result Determines odd Khovanov homology of 3-strand torus links.
We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links wi…
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…
We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.
Two graph homologies help compute embedding space.
problem Computing the rational homotopy group of long embeddings.
method Invented two graph homologies and constructed a map between them.
result A monomorphism from top hairy graph homology to top BCR graph homology.
We compute the -equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that for Seifert integral homology three-spheres. We show that the Manolescu invari…
Approaches for approximating persistent homology for large datasets.
problem Inability to compute persistent homology for large datasets.
method Multiple subsampling framework for statistical approximation of persistent homology.
result Derivation of finite sample convergence rates for empirical means of persistent homology.
Simplified computation of symmetric gl_1 homology for links.
problem Computing the symmetric gl_1 homology for uncolored links.
method Down-to-earth description, basis construction, and algorithm for computation.
result An algorithm and program for computing the invariant for uncolored links.
We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many…
We consider a family of tight contact structures on the three-dimensional torus and we compute the relative Contact Homology by using the variational theory of critical points at infinity. We will also show some algebraic equivariant homology reductions.
Skein theory aids in computing YB homology and cohomology groups.
problem Computing Yang-Baxter homology and cohomology groups.
method Skein theoretic approach to decompose the YB operator and its differentials.
result Computed second and third YB homology groups, confirming a conjecture.
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
In this paper we compute the singular homology of the space of immersions of the circle into the -sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together wi…
Study HOMFLY-PT homology structure for knots up to 11 crossings.
problem Understanding the structure of HOMFLY-PT homology for knots.
method Using Nakagane and Sano's knot data and the action, compute HOMFLY-PT -invariant and compare to invariants.
result Computed HOMFLY-PT -invariant for all knots in the dataset.
We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl_3-hom…
We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces t…
This paper computes the second quandle homology group of knot n-quandles.
problem Characterizing knots using quandle homology groups.
method Computation of the second quandle homology group for knot n-quandles.
result The second quandle homology group of knot n-quandles provides more information than the knot quandle's homology group.
In this article we study the Heegaard Floer link homology of -torus links. The Alexander multigradings which support non-trivial homology form a string of unit hypercubes in , and we compute the ranks and gradings of the homology in nearly all Alexander gradings. We also conjecture a compl…
In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over \Q an…
Higher hairy graph homologymath.AT
We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…
Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.