Invariants for connected sums of 3-manifolds, proving Furuta's theorem.
problem Computing and comparing Manolescu invariants of connected sums.
method Using inequalities and monopole theory to compute invariants.
result Proof of Furuta's Theorem using Manolescu invariants.
Estimates Manolescu's κ-invariant using spin 4-orbifolds.
problem Estimating κ-invariant of rational homology 3-spheres.
method Using spin 4-orbifolds bounded by rational homology 3-spheres.
result Restricts and determines the value of κ for specific cases.
Study constraints on knot surgery invariants using Seiberg-Witten theory.
problem Understanding invariants of knots in the three-sphere.
method Use Manolescu correction terms and Seiberg-Witten theory.
result Constraints on invariants for knots in the three-sphere.
In this paper, we study Manolescu's construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on 4-manifolds with boundary. The main goal is to introduce a new gauge fixing condition in order to apply the finite dimensional approximation technique. We also hope to provide a framewor…
New link homology theories for R P 3 \mathbb{RP}^3 RP 3 yield distinct invariants.
problem Extending link homologies for links in R P 3 \mathbb{RP}^3 RP 3 . method Introducing new Lee and Bar-Natan spectral sequences for links in R P 3 \mathbb{RP}^3 RP 3 . result New Lee and Bar-Natan invariants are distinct from existing ones.
If the Bing double of a knot K is slice, then K is algebraically slice. In addition, Heegaard--Floer concordance invariants developed by Ozsvath-Szabo and by Manolescu-Owens vanish on K.
Computes Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds.
problem Computing Pin(2)-equivariant monopole Floer homology for specific 3-manifolds.
method Uses Heegaard Floer/monopole Floer lattice complex and ranks of monopole Floer homology groups.
result Pin(2)-equivariant monopole Floer homology can be determined by ranks of monopole Floer homology groups.
New formula and properties of inverted Habiro series derived from GM series.
problem Understanding and manipulating knot invariants using series expansions.
method Developed a new formula for the inverted Habiro series (IHS) in terms of GM series and theta functions. Proved a multiplication formula for IHS.
result Established a natural ring structure for IHS and studied its residues, applying them to Dehn surgery formulas.
fkcompute calculates a knot invariant from a braid presentation.
problem Computing the Gukov-Manolescu invariant for complex knots and links.
method Three-phase pipeline: braid presentation search, state space encoding, R-matrix multiplication.
result fkcompute efficiently computes the invariant for knots and links up to 12 crossings.
New invariants defined for 3-manifolds, linking knot surgeries to polynomial relations.
problem Defining and studying new invariants for 3-manifolds based on root systems.
method Defined Z ^ G \hat{Z}^G Z ^ G and F K G F_K^G F K G for specific cases, derived recurrence relations. result Found a connection between knot surgeries and polynomial relations for symmetric representations.
New invariants defined for 3-manifolds, generalizing existing work.
problem Defining new invariants for 3-manifolds.
method Defining unfolded Seiberg-Witten Floer spectra for closed oriented 3-manifolds.
result Invariants generalize existing work and are computable for some examples.
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.
Study rules out exotic S 4 S^4 S 4 and # n C P 2 \# n \mathbb{CP}^2 # n CP 2 construction using zero surgery homeomorphisms.
problem Tackles the possibility of constructing exotic S 4 S^4 S 4 or # n C P 2 \# n \mathbb{CP}^2 # n CP 2 using zero surgery homeomorphisms. method Uses zero surgery homeomorphisms to relate slice properties of knots stably after a connected sum with a 4-manifold.
result Rules out the possibility of constructing exotic S 4 S^4 S 4 or # n C P 2 \# n \mathbb{CP}^2 # n CP 2 using zero surgery homeomorphisms. New invariant recovers known contact element and considers finite coverings.
problem Defining and studying new contact invariants in Seiberg-Witten Floer spectra.
method Cohomotopy set of Seiberg-Witten Floer spectrum, equivariant Borel cohomology.
result New invariant recovers known contact element and considers finite coverings.
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.
Study monopole h-invariants from a topological viewpoint.
problem Understanding the h-invariants of 3-manifolds.
method Using Lidman and Manolescu's description of monopole Floer homology.
result Prove several properties of the h-invariants.
We compute the Khovanov lasagna module of S²×S², confirming a conjecture.
problem Computing the Khovanov lasagna module of S²×S².
method Interpreting Manolescu-Neithalath's formula as a homotopy colimit, using categorified projectors.
result The Khovanov lasagna module of S²×S² is trivial.
Computes Pin(2)-equivariant Seiberg-Witten Floer homology for Seifert fibrations.
problem Homology cobordism between Seifert fiber spaces and integral homology spheres.
method Uses Heegaard Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Proves Manolescu's conjecture and identifies new homology cobordism obstructions.
New theorem connects tangle complements in 3-manifolds via Floer homology.
problem Understanding tangle complements in arbitrary 3-manifolds.
method Adapting holomorphic polygon counts to bordered-sutured Floer homology.
result Exact triangle relation for tangle complements in arbitrary 3-manifolds.
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.
New bounds on slice genus from knot invariants.
problem Bounding slice genus of knots in R P 3 \mathbb{RP}^3 RP 3 . method Using s s s -invariant to establish lower bounds. result Proves conjecture on slice genus bounds.
New invariant connects knot homology and BPS series for plumbed knot complements.
problem Understanding invariants of plumbed knot complements.
method Introducing an invariant unifying knot lattice homology and BPS series, proving a surgery formula.
result Proved a surgery formula relating the new invariant to the weighted graded root of the surgered 3-manifold.
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.
New homology theory for instantons, defined and analyzed.
problem Understanding instanton structures in symplectic geometry.
method Definition and analysis of twisted Symplectic Instanton homology, fitting into Floer Field theory.
result Properties of the new homology invariant for connected sums and Dehn surgery.
Study series invariants for plumbed 3-manifolds and their properties.
problem Understanding series invariants for plumbed 3-manifolds and their applications.
method Twisted root lattice, gluing and splitting properties, explicit description of lens spaces and Brieskorn spheres.
result Series verify gluing and splitting properties of 3-manifolds.
Researchers create a new invariant for knot theory.
problem Constructing invariants for knot theory.
method Extending a result to all irreducible representations of s l 3 \mathfrak{sl}_3 sl 3 . result Existence of a new invariant F K s l 3 F_K^{\mathfrak{sl}_3} F K sl 3 for any positive braid knot K K K . We generalize the Manolescu-Owens smooth concordance invariant delta(K) of knots K in the 3-sphere to invariants delta_{p^n}(K) obtained by considering covers of order p^n, with p prime. Our main result shows that for any odd prime p, the direct sum of delta_{p^n} as n ranges through the natural numbers, yields a homom…
Researchers determine quantum filtration structure of torus links.
problem Quantum filtration structure of torus links.
method Determined quantum filtration structure of Lee homology of torus links.
result Confirmed conjectures about torus links and s-invariant.
Study series invariants of plumbed 3-manifolds using root lattices.
problem Understanding invariants of plumbed 3-manifolds twisted by root lattices.
method Use formal series to study invariants, decompose Z ^ ( q ) \widehat{Z}(q) Z ( q ) , and compute in specific cases. result Show that Z ^ ( q ) \widehat{Z}(q) Z ( q ) is unique and decomposes into related series invariant under five Neumann moves. Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.
problem Constraints on smooth families of 4-manifolds with boundary.
method Use Manolescu's Seiberg-Witten Floer stable homotopy type.
result Inclusion map between diffeomorphisms and homeomorphisms is not a weak homotopy equivalence.
Proves generic independence and additivity of SL(2,C) Casson-Lin invariant.
problem SL(2,C) Casson-Lin invariant's parameter dependence and additivity under knot sums.
method Topology, microlocal analysis, algebraic geometry, Behrend functions.
result Generically independent and additive invariant under knot sums.
Researchers describe a new method to compute Seiberg-Witten-Floer spectra for a specific class of manifolds.
problem Computing Seiberg-Witten-Floer spectra for a specific class of manifolds.
method Using lattice homology, they provide an explicit combinatorial description of the spectra.
result They calculate Manolescu's κ-invariant for certain connected sums of the spaces.
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
New invariants for knotted surfaces derived from link homology.
problem Distinguishing knotted surfaces from unknotted ones.
method Link homology and A ∞ A_\infty A ∞ -algebras. result Invariant distinguishes unknotted sphere from certain knotted spheres.
Study knot Floer homology for connected sums, proving a formula and constructing a homomorphism.
problem Understanding knot Floer homology for connected sums of knots.
method Proved a formula for the conjugation action on knot Floer complexes of connected sums, constructed a homomorphism from smooth concordance group.
result Computed involutive invariants on large surgeries of connected sums of knots using the connected sum formula.
Study constraints on singular points of rational cuspidal curves using Heegaard Floer theory.
problem Constraints on singular points of rational cuspidal curves.
method Involutive Heegaard Floer homology theory.
result Results do not apply to rational cuspidal curves of even degree.
Study knot invariants to deduce Hopf invariant and propose a slope conjecture.
problem Understanding the topological significance of knot invariants and their relations.
method Analyzing the Gukov-Manolescu knot series and its coefficients, relating to Hopf invariant and colored Jones polynomials.
result Explicit formula for the Hopf invariant in terms of colored Jones polynomials for fibered knots up to 12 crossings.
The paper defines and computes a knot complement invariant for simple links.
problem Defining and computing a knot complement invariant for simple links.
method Using the large color R-matrix to study the Gukov-Manolescu series.
result Presentation of strange identities for positive braid knots.
Study contact surgeries on Legendrian links, proving invariant vanishing and overtwistedness.
problem Understanding contact surgeries on Legendrian links and their properties.
method Algebraic methods using Heegaard Floer homology and contact-geometric arguments.
result Vanishing of contact Ozsváth-Szabó invariant and overtwistedness of contact 3-manifolds.
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, …
New indefinite false theta functions match homological blocks for a specific 3-manifold.
problem Constructing homological blocks for certain 3-manifolds.
method Developing indefinite false theta functions and proving their equivalence to homological blocks.
result Indefinite false theta functions match homological blocks for the Poincaré homology sphere.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d d d -invariants with Froyshov-type inequality. Innovative series invariant for knot complements, linking to existing invariants.
problem Developing a new series invariant for knot complements.
method Introducing a three-variable series F K ( y , z , q ) F_K(y,z,q) F K ( y , z , q ) for plumbed knot complements. result Deriving a surgery formula relating F K ( y , z , q ) F_K(y,z,q) F K ( y , z , q ) to Z ^ ( q ) \hat{Z}(q) Z ^ ( q ) invariant. Using a combinatorial approach described in a recent paper of Manolescu, Ozsváth, and Sarkar we compute the Heegaard-Floer knot homology of all knots with at most 12 crossings as well as the τ τ τ invariant for knots through 11 crossings. We review the basic construction of \cite{MOS}, giving two examples that can be wor…
New local equivalence groups refine Rasmussen's s-invariant.
problem Refining Rasmussen's s-invariant for knot concordance.
method Introducing local equivalence groups combining Khovanov homologies.
result A refined s-invariant that determines triviality of knot images.
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.
Study computes SL(2,C) Floer cohomology for surgeries on knots.
problem Computing SL(2,C) Floer cohomology for knot surgeries.
method Sheaf-theoretic SL(2,C) Floer cohomology HP(Y) and HP_#(Y) relationships with SL(2,C) Casson invariant.
result Computes HP and HP_# for surgeries on two-bridge knots and non-small knots.
Paper constructs new equivariant Floer cohomology and proves invariance properties.
problem Invariance properties of spectral sequences in symplectic Khovanov homology and Heegaard Floer homology.
method Flexible construction of equivariant Floer cohomology with finite group action.
result Proves invariance properties of spectral sequences and introduces new concordance homomorphism.