Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…
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48 results for “Sato-Levine invariant”
The Dold-Whitney theorem helps classify bundles and gives a mod 4 slice obstruction.
problem Classifying -bundles and determining sliceability of links.
method Using the Dold-Whitney theorem to classify bundles and relate to the Sato-Levine invariant.
result The Dold-Whitney theorem's mod 4 obstruction coincides with the Sato-Levine invariant.
New method detects non-cobordant surface-links undetectable by other invariants.
problem Detecting non-cobordant surface-links using various invariants.
method A novel approach to distinguish non-cobordant surface-links.
result A new pair of non-cobordant surface-links cannot be distinguished by existing invariants.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
The intersection of three spheres in a sphere and a new application of the Sato-Levine invariantmath.GT
Study on intersections of three spheres in a larger sphere using Sato-Levine invariant.
problem Analyzing intersections of three 4-spheres within a 6-sphere.
method Transverse immersions and properties of intersections; Sato-Levine invariant.
result Equality β(L_1) + β(L_2) + β(L_3) = 0 for semi-boundary links.
We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previous…
We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's invariant. These invariants are seen to be naturally r…
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide ob…
Study on link concordance groups using Thom-Pontryagin construction.
problem Understanding the structure of link concordance groups through filtrations.
method Utilized Thom-Pontryagin construction and Sato-Levine invariants to analyze link concordance groups.
result The quotient group is isomorphic to .
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new qua…
The first part of this paper completes the classification of Whitney towers in the 4-ball that was started in three related papers. We provide an algebraic framework allowing the computations of the graded groups associated to geometric filtrations of classical link concordance by order n (twisted) Whitney towers in th…
Link concordance and Whitney towers linked to Milnor invariants.
problem Link concordance and Whitney towers classification.
method Clasper surgeries, Whitney towers, and Milnor invariants.
result Link concordance and Whitney towers classified in terms of Milnor invariants.
Study Heegaard Floer homology for two-component L-space links with zero linking number.
problem Characterize and calculate invariants for two-component L-space links with zero linking number.
method Use Heegaard Floer homology, Sato-Levine invariant, and Casson invariant to describe the relationship between these invariants and surgeries on links.
result Explicitly describe the relationship between the -function, Sato-Levine invariant, and Casson invariant for two-component L-space links with zero linking number.
In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant …
It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for some (piecewise-linear) links u…
Positive Linksmath.GT
Given a link L in the 3-sphere, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply-connected positive-definite smooth 4-manifold; the knot case has been studied extensively in work of Cochran-Harvey-Horn. Such a 4-manifold is necessarily homeomorphic to a (punctured) c…
This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) we…
Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensionscond-mat.str-el
Discovering topological quantum field theories in 2+1 and 3+1 dimensions.
problem Exploring topological orders in condensed matter lattice models.
method Calculating braiding statistics and link invariants of anyon excitations.
result Identifying new spin topological quantum field theories with specific knot/link invariants.
We prove the following results (1) (2) (3) on relations between -links and their components. (1) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Then we have Arf L=Arf L_1+Arf L_2. (2) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Then we have σL=σL_1+σL_2. (3) Let n\geq1. Then there is a nonribbon n-link L=(L_1, L…
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.
New equivalence found between knot invariants.
problem Understanding relationships between knot invariants.
method Comparing tree reductions of Kontsevich invariant with Orr invariants.
result Orr invariant of degree k is equivalent to tree reduction of Kontsevich invariant of degree <2k.
The invariant encompasses the Rozansky-Overbay invariant.
problem None explicitly stated in the abstract.
method Generalization of the Rozansky-Overbay invariant using the invariant.
result The invariant recovers the Rozansky-Overbay invariant.
New transverse invariants derived from Khovanov-type homology.
problem Transverse invariants from Khovanov-type homologies.
method Deformations of Khovanov homology to derive -invariants.
result Extracted -invariants from -invariants, proving vanishing criteria.
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 Milnor invariants for covering links to distinguish certain links.
problem Distinguish links using Milnor invariants for covering links.
method Generalize Hartley and Murasugi's covering linkage invariants and use cobordism invariants.
result First non-vanishing Milnor invariants of a Brunnian link modulo 2 equals a sum of linking numbers of covering links.
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.
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.
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces -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.
New invariant of virtual knots generalizes existing polynomial invariants.
problem No specific problem stated; focuses on a new invariant.
method Described a new transcendental function invariant of virtual knots.
result Generalizes several polynomial invariants of virtual knots.
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.
The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariantmath.GT
We show that the perturbative invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra , i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
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.
New invariant for alternating links is stronger than existing invariants.
problem Developing a stronger invariant for alternating links.
method Introducing invariant as an array of two-variable polynomials.
result The 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 for knots in the 3-sphere using Donaldson invariants and Floer's instanton homology.
result The invariant 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.
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.
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.
Smoothed Invariantsmath.GT
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…
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.
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 -invariant. We present a simpler new proof (in part) that the -invariant is ergodic. The -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.
Constructs BCOV invariant for Calabi-Yau pairs.
problem No specific problem stated; focuses on construction.
method Constructs BCOV invariant for Calabi-Yau pairs, covering classical and equivariant cases.
result Expected well-behaved under birational equivalence.
New link invariants from colored link polynomial.
problem Constructing stronger link invariants.
method Using skein invariant of colored links.
result Invariants stronger than HOMFLYPT polynomial.
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.
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 defines new invariants from Khovanov homology, linking them to existing ones.
problem Developing new transverse link invariants from Khovanov homology.
method Using Mackaay-Vaz approach to universal -homology, defining -invariants.
result Established relationship between new invariants and existing ones like Plamenevskaya's and Wu's.