A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Introduces combinatorial tangle Floer homology and its strand diagram equivalent.
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…
RNNs trained on spatial tasks develop grid-like spatial representations.
New method computes knot Floer homology for satellite knots.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
GridPyM handles grid diagrams for knot theory.
New method finds grid diagrams for many fibered knots.
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …
Half grid diagrams prove every link can be represented by a special type of grid diagram.
We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…
Minimal grid diagrams for 12-crossing prime knots identified.
New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.
New algebraic method for knot Floer homology computation.
Grid diagrams define invariants for knots in lens spaces.
Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.
Defines singular grid diagrams for various types of links.
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…
Real bordered Floer homology computes 3-manifolds with involution.
We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.
New diagonal knots found with non-torus structure.
We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3-manifolds and 4-manifolds …
Holomorphic polygons help calculate link complements' Floer homology.
We use grid diagrams to investigate the Ozsvath-Szabo concordance invariant tau, and to prove that |tau(K_1)-tau(K_2)|<=g, whenever there is a genus g knot cobordism joining K_1 to K_2. This leads to an entirely grid diagram-based proof of Kronheimer-Mrowka's theorem, formerly known as the Milnor conjecture.
Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.
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…
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.…
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…
Minimal grid diagrams found for 13-crossing prime knots.
Innovative contact invariant derived from Heegaard Floer homology.
Survey of knot Floer homology and bordered algebra techniques.
Combinatorial proof for knot Floer homology in branched covers.
Study finds new knot distances and chirally cosmetic bands using grid diagrams.
The paper extends Heegaard Floer homology to spatial graphs.
Develops equivariant grid homology for strongly invertible knots.
Researchers define new algebraic structures for knot Floer homology.
Permutations linked to knots and links, with unknots counted by Schröder numbers.
New proof confirms petal number for torus knots without modular condition.
In this short note we highlight some of the differences between cube diagrams and grid diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give some examples of links.
A method for vectorizing persistence diagrams simplifies topological data analysis.
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…
In paper "A new twist on Lorenz links" (Journal of Topology 2(2009), 227-248) Joan Birman and Ilya Kofman prove the coincidence of the class of Lorenz links and the class of twisted links. The proof in that work is algebraic. We will identify this class in terms of grid diagrams and provide a transparent geometric argu…
Study on typical knots and links using grid diagrams, focusing on size, components, and writhe.
In this article we study the differential graded algebra (DGA) invariant associated to Legendrian knots in tight lens spaces. Given a grid number one diagram for a knot in L(p, q), we show how to construct a special Lagrangian diagram suitable for computing the DGA invariant for the Legendrian knot specified by the dia…
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, …
Listed 19,513 prime knots with arc index 12-16.
Isomorphism found between Floer homology and contact geometry.
In two previous papers, the author showed how to decompose the Khovanov homology of a link into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for is decomposed into the union of two tangles. Since Khovanov homol…
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…