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.
Innovative link grid homology invariant measures concordance.
problem Measuring concordance of links.
method Generalized Ozsváth-Szabó τ-invariant via filtered link grid homology.
result Invariant remains unchanged under strong concordance and provides lower bound for slice genus.
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.
The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.
problem Defining and constructing a resolution cube for knot Floer homology of singular links in lens spaces.
method Defining grid homologies for singular links in lens spaces and using them to construct a resolution cube.
result A complete description of singular knot theory in lens spaces and a signed combinatorial resolution cube for knot Floer homology.
Researchers find a Steenrod square for link Floer homology.
problem Computing the second Steenrod square for link Floer homology.
method Explicitly framing moduli spaces and constructing a framed 1-flow category.
result An algorithm for computing the second Steenrod square for all grid homology versions.
Extends knot concordance invariant to balanced spatial graphs using grid homology.
problem Defining a concordance invariant for balanced spatial graphs.
method Using grid homology to extend the invariant from knots to spatial graphs.
result The combinatorial Υ invariant is a concordance invariant for balanced spatial graphs. Defines real link Floer homology for specific types of links.
problem Developing a new homology theory for certain types of links.
method Combining real Heegaard Floer homology and real sutured Heegaard Floer homology, using real grid diagrams in S3. result Observes structural and property properties of strongly invertible knots.
GRID invariants block certain Lagrangian cobordisms in 3D.
problem Obstructing decomposable Lagrangian cobordisms in 3D.
method Filtered GRID invariants and link Floer homology.
result GRID invariants obstruct decomposable Lagrangian cobordisms.
For knots in S^3, the bi-graded hat version of knot Floer homology is defined over Z; however, for a link L in S^3 with #|L|=l>1, there are 2^{l-1} bi-graded hat versions of link Floer homology defined over Z, the multi-graded hat version of link Floer homology is only defined over F_2 from holomorphic considerations, …
Same homology via different sign assignments in link Floer theory.
problem Comparing sign assignments in link Floer homology.
method Comparison of sign assignments from two different constructions.
result Small modification of sign convention results in identical chain complexes.
Study transverse and Legendrian invariants of certain link cables using Floer homology.
problem Transverse and Legendrian invariants of specific link cables.
method Combining grid diagrams and Floer homology, using inclusion maps of grid complexes.
result Transverse and Legendrian invariants of certain link cables are zero for large q/p. We re-derive Manolescu's unoriented skein exact triangle for knot Floer homology over F_2 combinatorially using grid diagrams, and extend it to the case with Z coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, …
We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial. Consequently, we prove that the Maslov grading on the Seidel-Smith symplectic link invari…
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler c…
The paper extends Heegaard Floer homology to spatial graphs.
problem Tackling the homology of transverse spatial graphs.
method Defining graph grid diagrams and proving their equivalence under moves, constructing chain complexes and showing invariance.
result Alexander-type polynomials and torsion invariants for transverse spatial graphs.
New invariant stops certain types of geometric transformations.
problem Obstructing decomposable Lagrangian cobordisms.
method Defined an invariant for Legendrian links.
result Obstructs decomposable Lagrangian cobordisms.
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descr…
Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, ξ_\st), where ξ_\st is the standard, tight contact structure.…
Constructs a spectrum for knot Floer homology without holomorphic geometry.
problem Computing knot Floer homology without using holomorphic geometry.
method Combinatorial definition and inductive construction of models for moduli spaces.
result Conjectures that the filtered homotopy type of the spectrum is an invariant of the knot.
Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…
Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alterna…
Study how braid properties affect transverse link invariants.
problem Understanding the relationship between braid properties and transverse link invariants.
method Use grid diagrams and Dehornoy's braid ordering to analyze 3-braids and higher-index braids.
result Non-zero Heegaard Floer transverse invariant for right-veering 3-braids.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) t…
Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.
Grid homology theory for spatial graphs extends skein sequence.
problem No specific problem stated; focuses on extending a sequence.
method Defined grid homology theory for spatial graphs and extended skein sequence.
result Skein exact sequence extended to grid homology for spatial graphs.
Grid homology shows knot unknotting lower bound.
problem Knot unknotting number determination
method Grid homology analysis
result Torsion homology classes order bounds unknotting number
New obstructions found for Lagrangian cobordisms in knot Floer homology.
problem Existence of Lagrangian cobordisms in knot Floer homology.
method Functorial behavior of Legendrian invariants with respect to decomposable Lagrangian cobordisms.
result New computable obstructions to Lagrangian cobordisms.
Grid homology properties for MOY graphs studied.
problem Defining and studying properties of grid homology for MOY graphs.
method Defined grid homology from Harvey and O'Donnol's work. Studied properties using oriented skein relation, edge contraction, and parallel edge unification.
result Properties of grid homology for MOY graphs were studied and defined.
The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.
problem Understanding grid homology for spatial graphs with various types of edges.
method Developed grid homology for spatial graphs with cut edges and applied it to prove a Künneth formula for connected sums.
result A Künneth formula for knot Floer homology of connected sums is proven using grid homology.
In this paper we study the relation between two diagrammatic representations of links in lens spaces: the disk diagram and the grid diagram and we find how to pass from one to the other. We also investigate whether the HOMFLY-PT invariant and the Link Floer Homology are essential invariants, that is, we try to understa…
Combinatorial proof shows knot invariant in Lipshitz's grid homology.
problem Proving knot invariance in Lipshitz's grid homology.
method Purely combinatorial proof.
result Proves 'minus' version of Lipshitz's double-point enhanced grid homology is a knot invariant.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
Combinatorial proof of grid homology properties.
problem Properties of double-point enhanced grid homology.
method Purely combinatorial proof, extended to Z coefficients. result Skein exact sequence obeyed by grid homology.
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
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…
Half grid diagrams prove every link can be represented by a special type of grid diagram.
problem Representing links using grid diagrams and related invariants.
method Defining half grid diagrams and constructing canonical pairs, proving equivalence to Jones' construction, relating to classical link invariants.
result Established a new method to relate the oriented Thompson index to classical link invariants and provided bounds for knot invariants.
New knot invariant Υ defined without holomorphic theory.
problem Defining knot invariants without holomorphic theory.
method Developed grid homology theory to define Υ. result Υ is a well-defined knot invariant and provides a lower bound on unknotting number. We prove grid homology existence and compute lens space groups.
problem Existence and computation of grid homology in lens spaces.
method Combinatorial proof and Sage program for computation.
result Existence of sign refined grid homology and ∂Z2=0. We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.
We explore a somewhat unexpected connection between knot Floer homology and shellable posets, via grid diagrams. Given a grid presentation of a knot K inside S^3, we define a poset which has an associated chain complex whose homology is the knot Floer homology of K. We then prove that the closed intervals of this poset…
The paper examines linking numbers in grid models and finds polynomial moments.
problem Analyzing linking numbers in grid models.
method Examined linking numbers as a random variable on isotopy classes of 2-component links, computed moments and limits.
result The uth moment of the linking number is a polynomial in the grid size with degree d≤u, and all odd moments vanish. Proofs knot homology connected sums using grid complexes.
problem Proving Künneth formula for knot Floer homology of connected sums.
method Constructs a quasi-isomorphism of grid chain complexes.
result Functorial behavior of Legendrian and transverse invariants under connected sum.
New method finds grid diagrams for many fibered knots.
problem Detecting fibered knots using grid diagrams.
method Developed an efficient method to identify grid diagrams with unique maximal Alexander grading states.
result Found suitable grid diagrams for 5385 of 5397 fibered prime knots with crossing number ≤ 13.
Defines singular grid diagrams for various types of links.
problem Unified description of singular links and related objects.
method Definition of singular grid diagrams and classification of equivalence relations.
result Unified description of singular links and related objects.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in CP2 under certain conditions. The paper describes and analyzes a knot concordance invariant ε using grid homology.
problem Understanding and computing the knot concordance invariant ε.
method Combinatorial description and grid homology techniques.
result Computed values and properties of the invariant ε for specific knots.
Introduces combinatorial tangle Floer homology and its strand diagram equivalent.
problem Defining and comparing tangle Floer homology in two equivalent ways.
method Presented strand diagrams and bordered grid diagrams, and discussed their equivalence.
result Explicit computations carried out to illustrate the equivalence of the two definitions.