We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is isomorphi…
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48 results for “pillowcase Floer homology”
We define an elementary relatively graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless character varieties of 2-tangle decompositions of knots. Calculations for torus knots…
Study on instanton homology of pretzel knots and pillowcase Floer homology.
problem Investigating instanton knot homology and Floer homology for a family of pretzel knots.
method Analyzes the reduced singular instanton knot homology and computes bounding cochains in the pillowcase.
result Computed bounding cochains in the pillowcase, revealing a sharp experimental law and rigidity asymmetry.
The paper studies traceless SU(2) representations and instanton gradings for specific knot types.
problem Analyzing traceless SU(2) representations and instanton gradings for two-bridge and -torus knots.
method Representation-theoretic data assembly and verification; explicit computation of character varieties and spectral-flow gradings.
result Explicit characterization of traceless representations and instanton gradings for specific knot types.
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…
Extends Khovanov homology spectral sequence using Heegaard Floer homology.
problem Relating Khovanov homology and Heegaard Floer homology of branched double covers.
method Involutive Heegaard Floer homology, bordered Floer homology, surgery exact triangle.
result Establishes spectral sequence connecting Khovanov homology and Heegaard Floer homology.
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with -cables and -cables.
Knot Floer homology matches fixed point Floer for fibred knots.
problem Understanding the algebraic structure of knot Floer homology.
method Proved isomorphism between Knot Floer and fixed point Floer homologies.
result Knot Floer homology of fibred knots matches fixed point Floer homology.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
Paper classifies Heegaard Floer minimal knots in sutured manifolds.
problem Classifying minimal knots in sutured manifolds.
method Uses Heegaard Floer homology.
result Agrees with instanton Floer homology classification when applicable.
New colored knot Floer homology defined using infinite full twists.
problem Defining a new homology theory for knots.
method Defining colored knot Floer homology through colimit of link Floer homology with infinite full twists.
result Colored knot Floer homology is a module over the colored knot Floer homology of the unknot.
New algebraic method for knot Floer homology computation.
problem Computing knot Floer homology efficiently.
method Extending bordered Floer homology to partial knot projections and establishing a pairing result.
result Identification of knot Floer homology with its algebraic definition.
Link Floer homology detects split links.
problem Detecting split links in link Floer homology.
method Module structure on link Floer homology and sutured Floer homology results.
result Link Floer homology detects split links.
Lecture notes on Heegaard Floer homology for 3-manifolds and knots.
problem Understanding Heegaard Floer homology for 3-manifolds and knots.
method Explains Heegaard diagrams, Heegaard Floer homology, and knot Floer homology.
result Relationship between knot and 3-manifold invariants.
Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.
problem Rank inequalities in Heegaard Floer homology.
method Using Hanselman-Rasmussen-Watson's bordered Floer homology, we extend their proof to rational homology solid tori.
result We provide rank inequalities for Heegaard Floer homology.
The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomologymath.GT
For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factoriz…
Proves properties of instanton knot Floer homology and connected sum formula.
problem Properties of instanton knot Floer homology.
method General properties of sutured Floer theories, Heegaard Floer and monopole Floer settings.
result Connected sum formula for instanton knot Floer homology.
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
Lecture notes on Heegaard Floer homology for beginners.
problem Introducing Heegaard Floer homology to newcomers.
method Illustrates algebraic structures via grid homology, then defines Heegaard Floer homology geometrically.
result Defines and describes key properties of Heegaard Floer homology.
Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.
problem Detecting and classifying links based on knot Floer homology ranks.
method Analyzes knot Floer homology ranks to identify and classify links.
result Classifies links of the second smallest knot Floer homology.
Introduces Floer lasagna modules using link Floer homology.
problem No specific problem stated; focuses on new mathematical concept.
method Inspired by skein lasagna module, uses link Floer homology.
result Computes Floer lasagna modules for specific 4-manifolds.
Real Heegaard Floer Homologymath.GT
Real Heegaard Floer Homology extends Li's real monopole Floer homology.
problem Computing invariants of three-manifolds with symmetries.
method Real version of Lagrangian Floer homology, combinatorial computation.
result Euler characteristic of real Heegaard Floer homology is a knot invariant.
Study shows inequality in Floer homologies for 3-manifold covers.
problem Establishing an inequality between Heegaard Floer homologies of branched covers.
method Using Heegaard Floer homology, the study establishes an inequality for dimensions of homologies of 3-manifold covers.
result An inequality is established between the dimensions of Heegaard Floer homologies of branched covers.
New method for Lagrangian Floer homology groups using flow trees.
problem Computing equivariant Lagrangian Floer homology.
method Constructing and exploiting an A-infinity module structure on the Floer complex.
result Established constructions of equivariant Lagrangian Floer homology groups.
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.
Contact gluing maps are shown to be equivalent in sutured Floer homology.
problem Equivalence of contact gluing maps in sutured Floer homology.
method Established a conjecture by Zarev, showing the contact gluing map is equivalent to the sutured Floer homology gluing map.
result Contact gluing maps are equivalent in sutured Floer homology.
Formula for Heegaard Floer multicurves of double tangles from knot complements.
problem Computing Heegaard Floer multicurve invariants of double tangles.
method Using a simple formula derived from knot complements, comparing with Khovanov homology.
result New characterisation of L-space knots and first example of thin knot Floer homology.
Paper extends link Floer homology detection to almost braided links.
problem Detecting links that are nearly braided using link Floer homology.
method Inspired by nearly fibered knots, uses clasp-braids and topological characterisation.
result Khovanov homology detects Whitehead link and infinite families of links.
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.
Heegaard Floer homology study confirms composition maps match up to homotopy.
problem Verifying consistency of composition maps in Heegaard Floer homology.
method Using Auroux and Zemke's results, proving agreement up to homotopy.
result Proves consistency of composition maps in Heegaard Floer homology.
Study calculates instanton Floer homology for surgeries on L-space knots.
problem Computing the framed instanton Floer homology of surgeries on L-space knots.
method Used Baldwin-Sivek contact invariant and proved homogeneity of the invariant.
result Framed instanton Floer homology agrees with Heegaard Floer homology for L-space knots.
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition, and for Seifert fibered integer homology spheres this is well known. We prove t…
For any link of two components in an integral homology sphere, we define an instanton Floer homology whose Euler characteristic is the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links i…
Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.
problem Understanding the action of symmetries on knot Floer homology.
method Relating knot Floer homology to Heegaard Floer homology via equivariant surgeries.
result Identify the action of the involution on Heegaard Floer homology with an action on knot Floer homology.
Cabling explained using curves on a punctured torus.
problem Understanding knot Floer homology through cabling.
method Interpreting Heegaard Floer homology via immersed curves.
result Formula for behavior of curves under cabling.
Framed Floer Homologymath.GT
For any three-manifold presented as surgery on a framed link (L,Λ) in an integral homology sphere, Manolescu and Ozsváth construct a hypercube of chain complexes whose homology calculates the Heegaard Floer homology of Λ-framed surgery on Y. This carries a natural filtration that exists on any hypercube of chain comple…
We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explic…
Injective maps in various link homologies induced by ribbon concordances proved.
problem Understanding which aspects of link homologies make them injective under ribbon concordances.
method Simple proof in a general setting for various homologies.
result Injective maps in knot Floer, Khovanov-Rozansky, and conic strong Khovanov-Floer homologies induced by ribbon concordances.
Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…
Proves a new skein exact triangle for real monopole Floer homology.
problem None explicitly stated; focuses on proving a new mathematical structure.
method Introduces a new exact triangle for real monopole Floer homology.
result Proves an unoriented skein exact triangle for real monopole Floer homology.
New method connects knot Floer homology with bordered Floer homology.
problem Computing involutive knot Floer homology of satellites.
method Invariant splitting principles for knot Floer complexes and bordered Floer homology.
result Involutive knot Floer homology of satellites can be computed from their companions.
Study improves Heegaard Floer homology relations for knots in homology spheres.
problem Improving relations in Heegaard Floer homology for knots in homology spheres.
method Proved inequality for d-invariants, used reduced Floer homology rank relations.
result Degree one maps between aspherical Seifert homology spheres are homotopic to homeomorphisms if Heegaard Floer homologies are isomorphic.
We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer hom…
Introduces integer-valued Heegaard Floer theory with canonical orientations.
problem Defining and proving properties of Heegaard Floer homology over integers.
method Using canonical orientations from coupled Spin structures, proving naturality and surgery exact triangle.
result Established integer-valued Heegaard Floer theory and proved its properties.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.