Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
Defines knot concordance invariant using instanton homology and Donaldson invariants.
problem Knot concordance and its classification.
method Defines an invariant φ for knots in the 3-sphere using Donaldson invariants and Floer's instanton homology. result The invariant φ coincides with a special case of an invariant defined by Froyshov. The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.
problem Understanding the homology of orbifolds.
method Constructing invariant and coinvariant Morse chain complexes for orbifolds.
result The homology of coinvariant Morse complexes computes the singular homology of the underlying space.
Khovanov homology invariant under Conway mutation.
problem Invariance of Khovanov homology under specific transformations.
method Strong geography restrictions and homological mirror symmetry.
result Classification of components of a Khovanov multicurve invariant.
Extends knot concordance invariant to balanced spatial graphs using grid homology.
problem Defining a concordance invariant for balanced spatial graphs.
method Using grid homology to extend the invariant from knots to spatial graphs.
result The combinatorial Υ invariant is a concordance invariant for balanced spatial graphs. New knot invariants discovered using mirror symmetry.
problem Categorify knot homology groups and explain their meaning.
method Homological mirror symmetry for new families of manifolds.
result Explicitly computable invariants for any Lie algebra.
The paper extends a knot invariant to graphs and connects it to homology cylinders.
problem Understanding the structure of homology cobordism groups.
method Using tangle Floer homology, the authors define a new invariant for embedded graphs and prove a concatenation formula.
result The new invariant induces a homomorphism on the homology cobordism group of homology cylinders.
Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
The paper examines lattice homology invariants of Seifert homology spheres.
problem Understanding homology cobordism invariants for Seifert fibered integral homology 3-spheres.
method Utilizes lattice homology and Heegaard Floer homology to study invariants.
result Reproves and extends the invariance of Seifert homology spheres' d-invariants and maximal monotone subroots. We define a link homology theory that is readily seen to be both isomorphic to reduced odd Khovanov homology and fully determined by data impervious to Conway mutation. This gives an elementary proof that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. We a…
Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.
problem Proving a conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres.
method Using Zemke's isomorphism between lattice and Heegaard Floer homology, the paper proves Némethi's conjecture.
result The conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres is proven.
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
New homology invariant for links in surfaces, a deformation of APS.
problem Defining a new homology invariant for links in surfaces.
method A 1-parameter family of homology invariants, motivated by instanton Floer homology.
result The new invariant recovers APS homology and has a stronger detection property.
The study examines invariants of homology cylinders and their relations to free nilpotent groups.
problem Understanding invariants of homology cylinders and their connections to free nilpotent groups.
method Extensions of Johnson homomorphisms, Milnor invariants, and Orr invariants of links to homology cylinders; establishment of a combined filtration.
result Determination of the image of the filtration under the invariants and investigation of relations among the invariants.
We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
New concordance invariant from spectral sequence on Khovanov homology.
problem Concordance invariants from spectral sequences.
method Constructing from E(−1) spectral sequence on Khovanov homology. result Provides a bound on the nonorientable slice genus.
Study s-invariants from sl(3)-link homology, extending to other characteristics.
problem Deepen understanding of s-invariants from Khovanov's homology. method Use Mackaay-Vaz universal sl(3)-link homology, extend to other characteristics, use scanning algorithm for calculations. result Extend s-invariants to other characteristics, particularly p=3. We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to st…
In this paper we answer the question posed by M.~Atiyah and give an explicit formula for Floer homology of Brieskorn homology spheres in terms of their branching sets over the 3--sphere. We further show how Floer homology is related to other invariants of knots and 3--manifolds, among which are the μˉ--invariant o…
New link homology theories for RP3 yield distinct invariants.
problem Extending link homologies for links in RP3. method Introducing new Lee and Bar-Natan spectral sequences for links in RP3. result New Lee and Bar-Natan invariants are distinct from existing ones.
We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relat…
The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants an…
New invariant fully describes finite type invariants of knots in homology 3-spheres.
problem Constructing a universal finite type invariant for knots in homology 3-spheres.
method Refined construction of a new invariant that is strictly stronger and universal.
result New invariant fully describes the graded space of finite type invariants of knots in homology 3-spheres.
Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain …
In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the f…
Study invariants of null-homologous knots in thickened surfaces.
problem Understanding concordance properties of knot invariants.
method Using Gordon-Litherland pairing and cobordism results for closed unoriented surfaces.
result Brown invariants are invariant under concordance of spanning surfaces.
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial.…
New invariant computed for Brieskorn homology spheres.
problem Computing an invariant for specific manifolds.
method Using Dijkgraaf--Witten invariants in topological K-theory.
result Computed invariant for Brieskorn homology spheres.
Khovanov homology invariant proved for links in RP3.
problem Proving Khovanov homology invariant for links in RP3. method Established functoriality of Khovanov homology for link cobordisms in IimesRP3. result Khovanov homology is an invariant of links in unparametrized RP3. Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 c…
New homology categorifies knotoid polynomial.
problem Generalizing knot theory to open-ended diagrams.
method Defined winding homology as a triply-graded invariant.
result Categorifies Turaev's polynomial invariant of knotoids.
New invariants derived from link homology for 4-manifolds.
problem Creating invariants for surfaces in smooth 4-manifolds.
method Skein lasagna modules based on equivariant and deformed glN link homology. result Non-vanishing and decomposition results for deformed glN skein lasagna modules. For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classe…
We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsváth and Szabó. This version has some clarifications and new figures.
Study of Khovanov homology invariants from U(1)imesU(1)-equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of U(1)imesU(1)-equivariant Khovanov homology using algebraic filtrations. result Extracted two families of concordance invariants.
M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behavi…
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
We prove that Khovanov homology and Lee homology with coefficients in F2 are invariant under component-preserving link mutations.
We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satisfying these axioms appear naturally in invariants of Reidemeister, Yoshikawa, and Roseman moves. Our homology has a degenerate subcomplex. T…
Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.
problem Cosmetic surgeries on knots in homology spheres and their constraints.
method Rational surgery formula of the Casson-Walker invariant for 2-component links.
result Constraints on knots and surgery slopes for cosmetic surgeries.
Defines an odd analog of Plamenevskaya's invariant for transverse links.
problem No specific problem stated; focuses on extending an invariant.
method Defines and analyzes an odd analog invariant in Khovanov homology.
result The odd analog invariant is an invariant of transverse links with similar properties.
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of K detects more structure of minimal genus Seifert surfaces for K. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homol…
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.