Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.
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This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the J…
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.
We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl3-link homology belo…
Morse-Conley-Floer Homologymath.DS
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…
Kasparov defined a distinguished K-homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K-homology of crossed product of C^{*}-algebras. Then we define the quantization of Hamiltonian G-spaces as a push-forward of the Dirac element. With this, we develop a K-…
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''.
We examine surgery on a knot in to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different app…
We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.
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 …
Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
Simplifies fixing Khovanov homology functoriality.
problem Fixing the sign indeterminacy in Khovanov homology.
method Adjusting signs of Reidemeister and Morse moves.
result Functoriality of Khovanov homology is fixed.
Study shows infinite-rank summand in homology concordance group of knots.
problem Homology concordance of knots in integer homology three-spheres.
method Knot Floer homology to construct homology concordance homomorphisms.
result Homology concordance group modulo knots from S^3 contains an infinite-rank summand.
Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes …
We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism.
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.
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …
Homological algebra used to study local equivalence of complex rings.
problem Local equivalence of bounded complexes over polynomial rings.
method Homological algebra approach
result Results have been proved in many places in the literature.
In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there…
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.
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
Refines quantum annular homology using stable homotopy methods.
problem Quantum annular homology lacks a stable homotopy refinement.
method Equivariant Burnside category approach, cyclic group action.
result Stable homotopy refinement of quantum annular homology constructed.
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…
We give a construction of a version of the Gromov-Witten classes, Q: H_*(J) -> HF_*(M) otimes ... otimes HF^*(M), within the context of symplectic Floer (co)homology. In particular, this gives a functorial approach to products and relations in symplectic Floer (co)homology.
The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for other Lie groups and representations. In particular, we introduce a new triply gr…
New approach connects pseudoisotopy theory to algebraic K-theory.
problem Understanding pseudoisotopy theory using algebraic K-theory.
method Constructing a zig-zag from pseudoisotopy space to algebraic K-theory space, using homological computations.
result Maps are p-locally (2n-4)-connected for large primes p.
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.
We define a "sutured topological quantum field theory", motivated by the study of sutured Floer homology of product 3-manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach t…
We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,ξ_std), we associate a corresponding multi-pointed nice Heegaard diagram. We then describe an explicit algorithm for computing the associ…
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…
Homology of abelian differentials stabilizes with more zeros.
problem Understanding the homology of abelian differentials with many simple zeros.
method Developed an h-principle for these strata, valid in a range of homological degrees.
result Homology stabilizes in a range where the number of simple zeros is large.
Mathematical pipeline identifies structural homology of knotted proteins.
problem Quantification and classification of protein structures, especially knotted proteins, require noise-free and complete data.
method Developed a geometric framework using persistent homology to analyze protein structures.
result Persistent homology accurately represents structural homology of knotted proteins and identifies geometric features of protein entanglement.
Study reveals striking uniformity in triply graded link homology for specific braids.
problem Investigating the structure of reduced triply graded link homology in specific degrees.
method Diagrammatic approach to Hochschild cohomology of Soergel bimodules.
result Homology often zero, especially in negative braid case, with striking uniformity.
Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds are reviewed. We present both in a unified approach using the representation of the Lie algebra of functions on itself by means of the hamiltonian vector fields. The use of the associated Lie algebroid allows to prove that the Lichnerowicz-Jacobi cohomolog…
New TQFT homologies help color graphs, potentially solving the four color theorem.
problem Graph coloring problem, especially the four color theorem.
method Topological quantum field theory (TQFT) to define homology theories.
result TQFT homologies can generate 4-face colorings of bridgeless planar graphs, offering a constructive approach to the four color theorem.
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spin structure isomorphic to its conjugate, we define the counterpart in this…
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.
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological st…
Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to…
Develops persistent Khovanov homology for tangles.
problem Lack of local topological features in evolutionary Khovanov homology for knots and links.
method Introduces a new mathematical framework using persistent Khovanov homology of tangles, employing functor and planar algebra.
result Provides a new method to characterize local features in curve-type data.
Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle, which complements the type D structure in a previous paper. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. T…
We use categorical annular evaluation to give a uniform construction of both and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield link homology, i.e. a link homology theory associated to the Lie superalge…
We generalize results of Lee, Gornik and Wu on the structure of deformed colored sl(N) link homologies to the case of non-generic deformations. To this end, we use foam technology to give a completely combinatorial construction of Wu's deformed colored sl(N) link homologies. By studying the underlying deformed higher r…
This paper explores adiabatic solutions of Haydys-Witten equations for knot homology.
problem Investigating instanton Floer homology and its relation to Khovanov homology.
method Analyzes decoupled Haydys-Witten equations and their equivalence to EBE solutions.
result Proposes an equivalence between adiabatic solutions of decoupled Haydys-Witten equations and non-vertical paths in EBE moduli space.
Discrete Morse theory simplifies Khovanov homology calculations.
problem Calculating Khovanov homology using traditional methods is complex.
method Employing discrete Morse theory for knot homologies.
result Advances understanding of Khovanov homology of 3-braids.
Quantum algorithm approximates Khovanov homology ranks.
problem Efficient computation of Khovanov homology ranks.
method Novel quantum algorithm with pre-thermalization procedure.
result Additive approximations to Khovanov homology ranks are hard problems.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.