Study topological 4-manifolds with specific fundamental groups.
problem Classify topological 4-manifolds with 4-dimensional fundamental group.
method Use algebraic topology, Poincaré duality, and Kirby-Siebenmann invariant.
result Two manifolds are homeomorphic if they are s-cobordant and have same Kirby-Siebenmann invariant.
Generalizes Rochlin's theorem to 4-manifolds with boundary and arbitrary 3-manifold boundaries.
problem Proving divisibility of self-intersection numbers in 4-manifolds with boundary.
method Combining Kirby-Siebenmann invariants, Brown invariants, and pin structures.
result Combinatorial description of Kirby-Siebenmann invariant for 4-manifolds with boundary.
We show that two closed, connected 4-manifolds with finite fundamental groups are CP2-stably homeomorphic if and only if their quadratic 2-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.
Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed…
Algorithm decides homeomorphism of 4-manifolds.
problem Deciding homeomorphism of closed, simply connected 4-manifolds.
method Represent 4-manifolds by Kirby diagrams, compute Kirby-Siebenmann invariant, and use algorithms for Kirby diagrams and intersection forms.
result Algorithmic decision for homeomorphism of 4-manifolds.
The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--b…
Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…
This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2.3) and the Hauptvermutung for manifolds.
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
The paper explores smooth equivariant rigidity and finds infinitely many exotic smooth structures.
problem Smooth equivariant rigidity for certain group actions on manifolds.
method Analysis of G-vector bundles and Atiyah-Singer index theorem. result Infinitely many exotic smooth structures for certain group actions.
The smoothing theory is revised to generalize to different disc embedding spaces.
problem Generalizing the smoothing theory to various disc embedding spaces.
method Revising the Morlet-Burghelea-Lashof-Kirby-Siebenmann theorem to apply to different versions of disc smooth embedding spaces.
result The delooping of disc embedding spaces is shown to be compatible with the Hatcher and Budney actions.
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
The θ invariant encompasses the Rozansky-Overbay invariant.
problem None explicitly stated in the abstract.
method Generalization of the Rozansky-Overbay invariant using the θ invariant. result The θ invariant recovers the Rozansky-Overbay invariant. Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.
problem Investigating invariants of 4-manifolds under group actions and Galois coverings.
method Functorial approach to equivariant invariants and study in Galois covering situations.
result Ordinary invariants of quotients are determined by equivariant invariants of the covering manifold.
The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.
problem Exploring gauge invariance of the Kuperberg invariant for specific 3-manifolds.
method Using hyperbolic 3-manifolds and finite-dimensional Hopf algebras.
result First examples of gauge invariants of general finite-dimensional Hopf algebras via topological methods.
New polynomial invariant distinguishes singular links.
problem Distinguishing singular links using existing invariants.
method Generalized quandle polynomial to singquandles and constructed a singular link invariant.
result New polynomial invariant distinguishes singular links with same counting invariant.
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 introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces J2+-invariant for oriented pairs of immersions, invariant under inverse tangencies and triple points. result Invariant changes under direct tangencies but remains invariant under orientation change and inverse tangencies.
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.
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
New invariant CWR for alternating links is stronger than existing invariants.
problem Developing a stronger invariant for alternating links.
method Introducing CWR invariant as an array of two-variable polynomials. result The CWR invariant is stronger than classical invariants like HOMFLYPT and Kauffman polynomials. 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. Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
New family of knots with epsilon invariant nonzero despite Upsilon and phi being zero.
problem Comparing smooth concordance invariants.
method Building an infinite family of knots.
result Found knots with epsilon invariant nonzero but Upsilon and phi zero.
Paper introduces a new invariant for virtual knotoids and proves it's a Vassiliev invariant of order one.
problem Tackles the problem of understanding invariants for virtual knotoids.
method Uses a 0-smoothing invariant constructed from local modifications at classical crossings.
result Demonstrates that the 0-smoothing invariant provides less information than the gluing invariant.
We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants…
As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree k is equivalent to the tree reduction of the Kontsevich invariant of degree <2k. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and …
Formula connects surface and curve invariants via slice transitions.
problem Computing surface invariants from curve invariants.
method Introducing differential measures for local changes across singular slice transitions.
result Explicit formula for surface invariant change during quadruple-point events.
In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called β-invar…
New concordance invariants phi and phi_j are defined and studied.
problem Understanding the relationships between different concordance invariants.
method Defined and analyzed new invariants phi and phi_j, and provided recursive formulas.
result Found infinitely many knots with specific combinations of zero and nonzero phi invariant.
We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic M-invariant. We present a simpler new proof (in part) that the M-invariant is ergodic. The M-invariant is a higher invariant, this means that for the magnetic field with closed magnetic lines the invariant is not a f…
Defines new link-homotopy invariants using Milnor's higher order link invariants.
problem Link-homotopy invariants for link maps of multiple components.
method Uses Milnor's higher order link invariants and combinatorial theory of cut-diagrams.
result Provides practical algorithms to compute these invariants and detects families of examples.
New invariants for singular knots and links defined using shadow structures.
problem Defining invariants for singular knots and links.
method Introducing action of singquandles on sets and defining shadow counting and polynomial invariants.
result Enhanced shadow counting invariant for singular knots and links.
We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, ρ-invariants and String-bordism invariants are derived as special cases. The main results are a secondary index theo…
Paper calculates L-invariant and L*-invariant for complex surface sums.
problem Calculating invariants for complex surface sums.
method Using pants complexes and dual curve complexes.
result First example of arbitrary large invariants for bridge numbers.
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.
The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invaria…
The study explores knot invariants using roots of unity.
problem Understanding knot invariants at roots of unity.
method Using Reshetikhin-Turaev method and generalizing ADO invariants.
result Clarified definitions and connections between different invariants.
Researchers prove birational invariance of BCOV invariant using motivic integration.
problem Proving the birational invariance of BCOV invariant for Calabi-Yau manifolds and varieties.
method Motivic integration theory applied to Calabi-Yau varieties with Kawamata log terminal singularities.
result Birational Calabi-Yau manifolds have the same BCOV invariant.
Power series invariant of hyperbolic 3-manifolds matches knot invariants.
problem Understanding topological invariants of hyperbolic 3-manifolds.
method Perturbative power series associated with ideally triangulated cusped hyperbolic 3-manifolds.
result The power series agrees with Kashaev and Andersen-Kashaev invariants to all orders.
New method to calculate 3-manifold invariants via skew-racks.
problem Calculating invariants of 3-manifolds.
method Introducing skew-racks with good involution and Property FR, defining cocycle invariants.
result Established new approach to obtain 3-manifold invariants via Dehn surgery.
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.
New invariant for knotted tori, similar to classical invariant.
problem Defining a new topological invariant for knotted tori.
method Analogous to Levine-Tristram invariant, using gauge theory for singular connections.
result Invariant matches Echeverria's invariant and Langte Ma's general result.
Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
problem Proving relations between Reshetikhin-Turaev and re-normalized link invariants for links.
method Analyzing higher order terms of re-normalized link invariants for plumbed links.
result Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
New relation found between 3-manifold Witt invariants and Z^-invariants.
problem Computing Witt invariants of 3-manifolds.
method Using Z^-invariants and analyzing homology spheres. result Alternative method to compute Witt invariants.