Study reveals weak knotting in confined polymers, not dominated by any single knot type.
problem Characterizing knotting in open, confined polymers.
method Modeling open curves as virtual knots, comparing lattice walks and ideal chains in confined and unconfined conditions.
result Weak knotting is a common feature in confined polymers, not dominated by any single knot type.
The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.
problem Classifying knot projections based on weak homotopy equivalence.
method Defining weak (1, 2, 3) homotopy and using it to find an invariant.
result There are an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections.
Functorial maps and weak parities are equivalent descriptions of rules of substitution virtual crossings for classical in diagrams of a knot in a way compatible with Reidemeister moves. We introduce the notion of maximal weak parity and describe it for knots in a given closed oriented surface. This weak parity defines …
Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.
problem Classifying knot projections under weak (1, 3) homotopy.
method Defines weak (1, 3) homotopy, introduces a map to knot isotopy classes, and determines trivial knots.
result Determines which knot projections are trivial under weak (1, 3) homotopy.
New homotopy types and invariants defined for knots.
problem Defining and characterizing different homotopy types of knot projections.
method Introducing strong and weak (1, 2) homotopies and defining new invariants.
result New necessary and sufficient conditions for homotopy equivalence of knot projections.
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic v…
The paper defines new homotopy relations on knot projections and classifies certain knot types.
problem Defining and classifying knot homotopy relations.
method Introducing cross chord numbers and using them to define strong and weak (1, 3) homotopies.
result Complete classification of knot projections with trivializing number two.
In this article, we prove that a tunnel number two knot induces a critical Heegaard splitting in its exterior if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Moreover, we prove that a connected sum of two 2-bridge knots or more generally that of two $(1,1)…
The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.
The study finds knots with specific surgeries that don't allow weak symplectic fillings.
problem Detecting weakly symplectic fillability of L-space knots after positive surgeries. method Analyzing arithmetic data from knot type and surgery coefficients to compute geometric invariants.
result Provides an infinite family of hyperbolic L-spaces that do not admit weakly symplectic fillings. Classifies crossings in tangles on surfaces, finding no nontrivial indices.
problem Classifying crossings in tangles on surfaces.
method Divides crossings into tribes compatible with Reidemeister moves and analyzes components, orders, and homotopy types.
result No nontrivial indices on classical knot diagrams.
The paper establishes conditions for link invariants to bound the weak splitting number.
problem Determining the minimum number of crossing changes to split a link into knots.
method Conditions on R-valued link invariants, link Floer homology, and known obstructions. result New bounds on the weak splitting number for many prime links.
For any Engel 4-fold, we show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the orbits of the Engel kernel. This is a relative, parametric and close h-principle.
New invariants lift Milnor invariants for 3-component links.
problem Classifying 3-component links using Milnor invariants.
method Defined and proved invariants γk(L), introduced h(L), and showed their equivalence. result Invariants γk(L) lift certain Milnor invariants and are invariant under weak cobordism. Constructs infinite families of hyperbolic knots satisfying a volume conjecture.
problem Constructing hyperbolic knots satisfying a volume conjecture.
method Dehn surgery methods to construct infinite families of hyperbolic knots.
result Obtained an explicit family of pseudo-Anosov mapping classes.
New index principle shows indistinguishability of certain knot crossings.
problem Identifying indistinguishable crossings in knot diagrams.
method Developed a universal index function on knot diagrams that respects Reidemeister moves and sign of crossings.
result Crossings of the same sign in a classical knot diagram cannot be distinguished by any inherent property.
A weak chord index Ind′ is constructed for self crossing points of virtual links. Then a new writhe polynomial W of virtual links is defined by using Ind′. W is a generalization of writhe polynomial defined in [6]. Based on W, three invariants of virtual links are constructed. These invariants can be used to …
Authors compute stable homology of torus knots using a new deformation technique.
problem Computing stable homology of torus knots.
method Link-splitting deformation (y-ification) of link homology.
result Explicit computation of y-ified glN stable Khovanov--Rozansky homology of torus knots. The Heegaard Floer correction term (d-invariant) is an invariant of rational homology 3-spheres equipped with a Spinc structure. In particular, the correction term of 1-surgeries along knots in S3 is a (2Z-valued) knot concordance invariant d1. In this paper, we estimate d1 for the (2,q)-cabl…
Let X be the exterior of connected sum of knots and Xi the exteriors of the individual knots. In \cite{morimoto1} Morimoto conjectured (originally for n=2) that g(X)<σi=1ng(Xi) if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of …
Goodwillie's model connects knot spaces to cosimplicial spaces, aiding in knot homotopy computation.
problem Computing the homotopy spectral sequence of the space of long knots.
method Using cosimplicial spaces and Goodwillie's correspondence, computing the first page of the spectral sequence.
result A combinatorial interpretation of differentials in the spectral sequence, via marked graphs.
Stability of knots at low regularity, and symmetric critical knots for Möbius energy.
problem Stability of knot equivalence at low regularity.
method Localized Gromov distortion and Hausdorff-distance criteria.
result Compactness theorem for knot equivalence classes and existence of symmetric critical knots for Möbius energy.
In the 60's Levine proved that if R is a slice knot, then on any genus g Seifert surface for R there is a g component link J, called a derivative of R, on which the Seifert form vanishes. Many subsequent obstructions to R being slice are given in terms of slice obstructions of J. Many of these obstructi…
We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in th…
The paper provides a new inequality for 4-manifolds and uses it to study knot sliceness and symplectic embeddings.
problem Understanding sliceness of knots and symplectic embeddings in 4-manifolds.
method An adjunction inequality for embedded surfaces in 4-manifolds with contact boundaries.
result Infinitely many knots are topologically H-slice but not smoothly H-slice in certain 4-manifolds.
A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The quotient of \widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the…
We define the {\it Wirtinger number} of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. …
The abstract introduces a new A∞ duality via LSFT algebra.
problem Legendrian knot duality and its A∞ extension. method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of A∞ bimodules over Aug+. result Explicit construction of homotopy inverse for the A∞ Sabloff map. The paper defines functions from spherical curves and chord diagrams, proving invariance under specific Reidemeister moves.
problem Defining and proving invariance of functions derived from spherical curves and chord diagrams.
method Introducing ∑iαixi and ∑iαiildexi functions, and defining relators of Type (I) ((SII), (WII), (SIII), or (WIII), resp.). result If ∑iαiildexi vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), resp.), then ∑iαixi is invariant under specific Reidemeister moves. We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we …
New structures defined for studying contact foliations and their geometry.
problem Understanding dynamics of contact foliations and their applications.
method Define and study weak nearly S- and weak nearly C-structures.
result Characterize weak nearly S- and weak nearly C- submanifolds in weak nearly Kähler manifolds.
Tensoring p-weak differentiable structures preserves their properties.
problem Tensorization of p-weak differentiable structures. method Proving the product of p-weak charts is a p-weak chart, and showing isometric embeddings. result Tensorization of p-weak differentiable structures is possible under certain conditions. We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expe…
The study examines conditions for weak nearly cosymplectic manifolds to split into products.
problem Understanding the curvature and topology of weak nearly cosymplectic manifolds.
method Analyzes the conditions for splitting and characterizes specific manifolds.
result Conditions for weak nearly cosymplectic manifolds to become Riemannian products are identified.
Subset selection improves weak supervision performance.
problem Optimizing the use of weakly-labeled data.
method Combining pretrained data representations with the cut statistic for subset selection.
result Subset selection improves weak supervision performance by up to 19%.
Study counts sub-chord diagrams to classify spherical curves.
problem Classifying spherical curves using chord diagrams.
method Counting sub-chord diagrams under specific moves.
result New invariant classifies prime reduced spherical curves.
Defines weak geodesics on specific subsets of manifolds.
problem Characterizing geodesics on prox-regular subsets of Riemannian manifolds.
method Defining weak geodesics as continuous curves with weak regularities, and characterizing them as viscosity critical points of the energy functional.
result Characterizes weak geodesics on prox-regular subsets of Riemannian manifolds.
Study weak conjugacy in surface homeomorphisms.
problem Understanding weak conjugacy in homeomorphisms of surfaces.
method Exploring the group of homeomorphisms isotopic to the identity.
result New insights into weak conjugacy relations.
New model shows weak teachers can help strong students learn even with imperfect labels.
problem Improving strong student's performance with weak teacher's imperfect pseudolabels.
method Stylized overparameterized spiked covariance model with Gaussian covariates, proving two phases of generalization.
result Provable successful and random guessing phases of strong student's generalization.
Introduces weak (p,k)-Dirac structures in geometric settings.
problem Defining and analyzing new geometric structures.
method Introducing and studying weak (p,k)-Dirac structures in TM⊕ΛpT∗M. result Weak (p,k)-Dirac structures contain more information than (p,k)-Lagrangian structures. RAVEN improves weak-to-strong generalization under distribution shifts.
problem Weak models fail to supervise strong models effectively under distribution shifts.
method RAVEN dynamically learns optimal combinations of weak models and strong model parameters.
result RAVEN outperforms existing methods by over 30% on out-of-distribution tasks.
Study weak f-K-contact manifolds, finding Einstein-type metrics and solitons.
problem Characterize and study geometric properties of weak f-K-contact manifolds. method Analyzing weak metric f-structures, using Killing vector fields, and Jacobi operators. result Einstein weak f-K-contact manifolds are Ricci flat. Study the geometry of weak para-f-structures and subclasses.
problem Understand the geometry of weak para-f-structures and their subclasses.
method Express covariant derivative of f, prove Killing characteristic vector fields, show foliations, and demonstrate rigidity.
result Prove that characteristic vector fields are Killing and ker f defines a totally geodesic foliation.
Weak labels can significantly speed up learning for strong tasks.
problem Learning with limited strong labels.
method Using weak labels to accelerate learning of strong tasks.
result Weak labels can accelerate learning to O(icefrac1n) rate. The study explores new metric structures on manifolds, linking them to Einstein metrics.
problem Characterizing and understanding weak K-contact manifolds and their properties.
method Analyzing weak K-contact manifolds and their properties, including the parallel Ricci tensor and generalized Ricci soliton structures.
result Sufficient conditions for weak K-contact manifolds with specific properties to be Einstein manifolds.
Study on Ricci solitons and Einstein metrics in weak β-Kenmotsu manifolds.
problem Characterizing Einstein metrics in weak β-Kenmotsu manifolds.
method Adapted ∗-Ricci tensor to weak almost contact manifolds and studied its interaction with weak β-Kenmotsu structures. result New characteristics of Einstein metrics obtained.
This paper studies properties of weak reducing pairs in critical Heegaard splittings.
problem Characterize weak reducing pairs in critical Heegaard splittings.
method Analyze the properties of weak reducing pairs in critical Heegaard splittings.
result Provide a necessary condition for a Heegaard surface to be critical.
We study weakened f-structures on manifolds, generalizing classical results.
problem Classical f-structures and their properties on manifolds. method Introduced and studied weakened f-structures, subclasses, and their properties. result Generalized known results on globally framed f-manifolds.