Knot concordance linked to involutive knot Floer homology equivalence.
problem Understanding knot concordance through algebraic topology.
method Involutive knot Floer homology and stable equivalence.
result Concordant knots have equivalent involutive knot Floer complexes.
Study shows how knot Floer homology and bordered Floer theory are linked.
problem Understanding the relationship between knot Floer homology and bordered Floer theory.
method Proves local equivalence and establishes algebraic formulas.
result Involutive knot Floer homology and bordered Floer theory determine each other under certain conditions.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
Corrected mistakes in knot Floer homology formulas for spliced knot complements.
problem Incorrect assumption about knot Floer homology involution.
method Considered the contribution from a possibly non-trivial involution.
result Corrected splicing formulas in knot Floer homology.
Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.
problem Understanding the homology cobordism group of 3-manifolds formed by knots and their mirrors.
method Using involutive Floer homology and gauge theory.
result Locally trivial involutive Floer homology on certain splices of knots and their mirrors.
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.
Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d and dˉ…
New invariants improve Heegaard Floer slice genus and clasp number bounds.
problem Improving bounds for knot concordance.
method Using knot Floer homology and involutive correction terms.
result Improved slice genus and clasp number bounds proved.
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.
New knot invariants from Floer theory help bound knot three-genus.
problem Bounding the three-genus of knots.
method Involutive Heegaard Floer knot theory.
result Two new concordance invariants defined.
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.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
Proves spectral sequence for real Heegaard Floer homology.
problem Real Heegaard Floer homology and its localization.
method Proves existence of a localization spectral sequence.
result Existence of spectral sequence for real Heegaard Floer hat variant.
Real Heegaard Floer homology gets a new grading for certain 3-manifolds.
problem Real Heegaard Floer homology groups get an absolute Z/2 grading under specific conditions.
method Analyzes real Heegaard Floer homology groups with an involution and nullhomologous fixed points.
result Defines a new invariant of knots equal to the Alexander polynomial evaluated at i.
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact t…
New knot invariants from covering involutions help deduce linear independence results.
problem Understanding the smooth concordance group of knots.
method Covering involutions and adapted ideas from previous research.
result Novel linear independence results in the smooth concordance group of knots.
New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn. The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.
problem Calculating numerical invariants for specific 3-manifolds.
method Involutive Heegaard Floer homology techniques and spin filling constraints.
result Established new constraints and obstructions for 3-manifolds.
We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…
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.
New invariants from Seiberg-Witten theory for 3-spheres with involution.
problem Equivariant Seiberg-Witten Floer theory of rational homology 3-spheres.
method Coupling involution to Seiberg-Witten theory, constructing delta-invariants.
result New Floer-theoretic invariants with properties and applications.
Formula derived for cabled knots' concordance invariants.
problem Understanding involutive concordance invariants of cabled knots.
method Proved a formula relating cabled knots' invariants to companion and pattern knots.
result Iterated cables of certain knots are not smoothly slice.
The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some invol…
Real Heegaard Floer theory shown to be natural and invariant.
problem Defining and proving naturality of real Heegaard Floer homology.
method Defined and proved naturality of real Heegaard Floer homology and other related theories.
result Real Heegaard Floer homology is shown to be natural and admits an action of the equivariant mapping class group.
Formula connects Lefschetz number to Frøyshov invariant and Murasugi signature.
problem Calculating the Lefschetz number for a specific map in Floer homology.
method Skein-theoretic argument and exact triangle in monopole Floer homology.
result Formula for Lefschetz number in terms of Murasugi signature and Frøyshov invariants.
The paper uses Heegaard Floer homology to study homology cobordism groups.
problem Understanding the structure of homology cobordism groups.
method Defined an invariant using Heegaard Floer homology, analogous to Stoffregen's connected Seiberg-Witten Floer homology.
result Computed involutive correction terms for certain three-manifolds.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
Study shows infinite-rank summand in homology cobordism group.
problem Identifying infinite-rank summands in homology cobordism groups.
method Introduced an algebraic variant of the involutive Heegaard Floer package.
result Demonstrated an infinite-rank summand in the homology cobordism group.
New bounds on nonorientable four-ball genus for torus knots.
problem Finding bounds on the nonorientable four-ball genus of torus knots.
method Combining knot Floer homology techniques to derive lower bounds.
result Sharp bounds for several families of torus knots, including T4n,(2n±1)2. We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involu…
The paper develops a new Floer theory for 3-manifolds with involutions.
problem Constructing a Floer theory for 3-manifolds with symmetries.
method Develops a framed real monopole Floer homology for 3-manifolds with involutions and provides definitions for invariants.
result Defines Z-valued framed real Seiberg--Witten invariants for 4-manifolds with involutions. Research examines triangulations and homology cobordism groups.
problem Understanding the three-dimensional homology cobordism group.
method Local equivalence methods from Pin(2)-equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
result Review of known results and new insights into the group.
Study homology cobordism and plumbed manifolds using Heegaard Floer homology.
problem Understanding homology cobordism and local equivalence between plumbed manifolds.
method Use involutive Heegaard Floer homology to analyze linear combinations of almost-rational plumbed three-manifolds.
result Proves that certain plumbed manifolds are not torsion in the homology cobordism group.
New invariant detects infinite order cabled knots.
problem Detecting infinite order knots in the concordance group.
method Involutive knot Floer homology and cabling techniques.
result Infinite family of knots with infinite order in concordance group.
Real bordered Floer homology computes 3-manifolds with involution.
problem Computing real Heegaard Floer homology for 3-manifolds with involution.
method Using bordered Heegaard Floer algebra modules, a practical algorithm is given.
result Computes real Heegaard Floer homology for real 3-manifolds with connected fixed set.
New real invariants for 3-manifolds and links.
problem Defining real structures in Floer homology.
method Introducing real spin-c structures and applying monopole Floer homology.
result Invariants for links via double branched covers.
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with d(Y)=d(Y)=d(Y). We also construct a homomorphism from the three-dimensional homolo…
We show that a certain involution on the well known homology sphere (the boundary of the Mazur manifold) induces a nontrivial homomorphism on its Heegard-Floer homology groups (recently defined by Ozsvath and Szabo). We discuss a possible application of this to constructing exotic smooth structures on 4-manifolds.
New invariants detect corks obstructing homology ball extensions.
problem Detecting corks obstructing homology ball extensions.
method Adapting Heegaard Floer homology formalism for local equivalence, using monotonicity and surgery.
result Established new families of corks and proved various examples are strongly non-extendable.
New invariants for 4-manifolds obstruct certain surface pairs.
problem Obstructing the existence of specific surface pairs in 4-manifolds.
method Defining new mixed invariants using involutive Heegaard Floer homology.
result The new invariants obstruct the existence of certain surface pairs.
We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link L in S3. We prove that there exists a spectral sequence of F[Q]/Q2-modules (where Q has degree −1) which converges to $\widetilde{\m…
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.
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 (n,nm)-cables and (2,2m)-cables. 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.
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.