arXiv research

Grid homology confirms the Upsilon invariant in knot theory.

Computes homology of an obstruction chain complex in grid homology.

Grid homology theory for spatial graphs extends skein sequence.

Grid homology shows knot unknotting lower bound.

Grid homology properties for MOY graphs studied.

Grid homology invariant proved for lens space links.

The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.

Extends knot concordance invariant to balanced spatial graphs using grid homology.

Develops equivariant grid homology for strongly invertible knots.

Combinatorial proof of grid homology properties.

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

Researchers find a Steenrod square for link Floer homology.

The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.

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…

Proofs knot homology connected sums using grid complexes.

New method finds grid diagrams for many fibered knots.

According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant $Υ$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that $Υ$, as introduced this way, is a well-defined knot invariant. We reprove some important proposit…

We prove that the "minus" version of Lipshitz's double-point enhanced grid homology is a knot invariant through purely combinatorial means.

The paper describes and analyzes a knot concordance invariant ε using grid homology.

GRID invariants block certain Lagrangian cobordisms in 3D.

Defines real link Floer homology for specific types of links.

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, …

Extends knot invariant to filtered grid complexes.

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…

Lecture notes on Heegaard Floer homology for beginners.

Constructs a spectrum for knot Floer homology without holomorphic geometry.

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, …

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Moti…

We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same tr…

We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topologic…

We present a combinatorial proof for the existence of the sign refined grid homology in lens spaces, and a self contained proof that $\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion…

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.…

Algorithm computes knot invariants for surgeries on prime knots.

We introduce a generalization of the Ozsváth-Szabó $τ$-invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus lin…

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…

Looking for an efficient algorithm for the computation of the homology groups of an algebraic set or even a semi-algebraic set is an important problem in the effective real algebraic geometry. Recently, Peter Burgisser, Felipe Cucker and Pierre Lairez wrote a paper [1], which made a step forward by giving an algorithm …

This paper is a short introduction to the combinatorial version of tangle Floer homology defined in "Combinatorial tangle Floer homology". There are two equivalent definitions---one in terms of strand diagrams, and one in terms of bordered grid diagrams. We present both, discuss the correspondence, and carry out some e…

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…

New invariant stops certain types of geometric transformations.

This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer…

The aim of this paper is to study the skein exact sequence for knot Floer homology. We prove precise graded version of this sequence, and also one using $\HFm$. Moreover, a complete argument is also given purely within the realm of grid diagrams.

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…

Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic branched cover $Σ_m(K)$ obtained from a grid diagram for $K$, we give a combinatorial proof for the invariance of the associated combinatorial knot Floer homology over $\mathbb{Z}$.

We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K in its m-fold cyclic branched cover Sigma^m(K), and we give computations when m=2 for over fifty three-bridge knots with up to eleven crossings.

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.

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the exist…

The study of tiling homology on flat surfaces, proving impossibility of certain tilings.

We study the Ozsváth-Szabó-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables $\mathscr{L}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables $\mathscr{L}_{p,q}$ are constructed from the grid diagram of $L$. The main result is $\hatθ(\mathscr{L}_{p,q})=0$ if and only…