Proves connection between ℓ2-Betti numbers and BNSR invariants.
problem Relationship between ℓ2-Betti numbers and BNSR invariants. method Analyzes ℓ2-Betti numbers and BNSR invariants of groups. result If the nth ℓ2-Betti number is non-zero, then the nth BNSR invariant over Q is empty. Study investigates lattices fibring over the circle, focusing on BNSR invariants.
problem Investigating BNSR invariants of irreducible uniform lattices.
method Examines BNSR invariants and Bestvina-Brady groups to understand lattice properties.
result Irreducibility of lattices is linked to the vanishing of BNSR invariants for all finite-index subgroups.
Researchers compute BNSR-invariants for surface Houghton groups.
problem Computing BNSR-invariants for surface Houghton groups.
method Proved Stein--Farley cube complex is CAT(0), adapted Zaremsky's method.
result Computed BNSR-invariants of surface Houghton groups.
The study bounds Dehn functions of coabelian subgroups using a second BNSR invariant.
problem Bounding the Dehn function of coabelian subgroups in hyperbolic groups.
method Using an area-radius pair for a finitely presented group and a second BNSR invariant.
result Finitely presented coabelian subgroups of hyperbolic groups have polynomially bounded Dehn functions.
Investigates BNSR invariants of link and knot groups, proving specific properties.
problem Characterizing finiteness properties of normal subgroups in link and knot groups.
method Analyzes BNSR invariants of link and knot groups, proving specific properties.
result Proves specific conditions for finiteness properties of link and knot groups.
The paper shows BNSR-invariants of McCool groups are either dense or empty.
problem Characterizing BNSR-invariants of McCool groups.
method Understanding higher generation properties of abelian subgroups and using a general criterion for characters.
result Every BNSR-invariant Σm of a McCool group is either dense or empty. The BNSR-invariants of a group G are a sequence Σ1(G)⊇Σ2(G)⊇⋯ of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free…
BNSR invariants are contained in the complement of tropical varieties.
problem Understanding the relationship between algebraic and topological invariants.
method Showed containment of BNSR invariants in the complement of tropical varieties.
result BNSR invariants are contained in the complement of tropical varieties.
The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.
problem Refining Alexander polynomials and bounds BNSR Σ-invariants for 3-manifolds and Kähler manifolds.
method Introduces twisted homology jump loci and uses tropical geometry to obtain bounds.
result Sharp bounds for BNSR Σ-invariants and obstructions to geometric realizability.
We inspect the BNSR-invariants Σm(Pn) of the pure braid groups Pn, using Morse theory. The BNS-invariants Σ1(Pn) were previously computed by Koban, McCammond and Meier. We prove that for any 3≤m≤n, the inclusion Σm−2(Pn)⊆Σm−3(Pn) is proper, but Σ∞(Pn)=Σn−2(Pn). We writ…
We give a complete computation of the BNSR-invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT(0) cube complex by their interse…
Bieri, Geoghegan and Kochloukova computed the BNSR-invariants Σm(F) of Thompson's group F for all m. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which F acts.
Generalizes results on Bieri-Neumann-Strebel-Renz invariants and tropical varieties.
problem Relationship between Bieri-Neumann-Strebel-Renz invariants and homology jump loci.
method Uses tropical varieties to detect components of homology jump loci and generalizes results to integral coefficients.
result Provides a better upper bound for Bieri-Neumann-Strebel-Renz invariants and classifies Kähler groups.
Complete description of BNSR invariants for Lodha-Moore groups, proving finiteness properties.
problem Computing Bieri-Neumann-Strebel-Renz invariants for Lodha-Moore groups.
method Variation of Bestvina-Brady discrete Morse theory applied to cluster complex.
result All higher invariants of Lodha-Moore groups coincide with the second invariant, proving finiteness properties.
This paper stems from the observation (arising from work of T. Delzant) that "most" Kähler groups virtually algebraically fiber, i.e. admit a finite index subgroup that maps onto Z with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, i.e. t…
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 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.
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 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…
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.