The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.
problem Characterizing isotopy classes of links in 3-manifolds, especially in contact structures.
method Developed theory of links transverse to a nowhere-zero vector field, constructing examples in both Legendrian and pseudo-Legendrian settings.
result Non-simple isotopy classes of links exist, including Legendrian and pseudo-Legendrian examples.
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to t…
We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x to have simple (convex and/or con…
The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples where Vassiliev in…
We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a 3-manifold M that are transverse to a nowhere-zero vector field V up to the corresponding isotopy relation. Such knots are called …
The paper calculates actions of string link operations for 4- and 5-component links.
problem Classifying link-homotopy classes of string links.
method Explicit calculation of actions of partial conjugations and conjugations for 4- and 5-component string links.
result Presentations of link-homotopy classes for 4- and 5-component links.
Study proves chainmail links are L-space links.
problem Understanding L-space links in chainmail links.
method Inspired by alternating links, uses flat augmentation.
result Proves alternating chainmail links are L-space links.
New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
Paper finds linking numbers for Montesinos links using a simple algorithm.
problem Calculating linking numbers for Montesinos links.
method Simple proof and numerical algorithm for rational links, extending to Montesinos links.
result Linking numbers found for any two components in Montesinos links.
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
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.
Extends positive and almost positive links to successively almost positive ones.
problem Extending properties of positive and almost positive diagrams and links.
method Introducing successively almost positive diagrams and links, and analyzing their properties.
result Improves known results of positive and almost positive links.
Study of knots and links in 2-complexes, defining linking numbers and polynomials.
problem Understanding knots and links in 2-dimensional complexes.
method Definition of linking numbers and Kauffman-type bracket polynomials for links in 2-complexes.
result Established relationships between 2-complexes and knots/links in 3-manifolds.
A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-…
We generalized the periodic links to \emph{transitive} links in a 3-manifold M. We find a complete classification theorem of transitive links in a 3-dimensional sphere R3. We study these links from several different aspects including polynomial invariants using the relation between link polynomials of…
Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
New link invariants from diagram colorings match link widths.
problem Defining link widths via diagram colorings.
method Colorings of link diagrams to define invariants and prove their equivalence to link widths.
result Invariants of link widths calculated algorithmically.
Innovates a three-component link homotopy invariant.
problem Classifying three-component link maps up to homotopy.
method Developed tools and invariants for distinguishing three-component link maps.
result Found three-component link maps that are not homotopic.
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…
New methods show hyperbolicity of Brunnian links.
problem Determining hyperbolicity of Brunnian links.
method Geometric/combinatorial topology techniques applied to links with at least one unknotted component.
result Determine hyperbolicity for all Brunnian links.
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link d…
The paper classifies links up to link-homotopy using claspers.
problem Classifying links up to link-homotopy.
method Using Habiro's clasper calculus, defining a linear representation of the homotopy braid group, and providing a geometric proof.
result Geometric proof of Levine's classification of 4-component links and further classification of 5-component links in the algebraically split case.
Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of 4-component links were classified by Levine with enormous algebraic computations. We modify the results by using Habiro's clasper theory. The new classification…
Link Floer homology detects split links.
problem Detecting split links in link Floer homology.
method Module structure on link Floer homology and sutured Floer homology results.
result Link Floer homology detects split links.
New link types derived from old links.
problem Understanding the structure of modular links.
method Analyzing the complements of arithmetic modular links and comparing them to augmented chainlinks.
result Arithmetic modular links are homeomorphic to augmented chainlinks.
Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…
This report lists the link diagrams in S^3 for all principal congruence link complements for which such a link diagram is known. Several unpublished link diagrams are included. Related to this, we also include one link diagram for an arithmetic regular tessellation link complement.
Extends Benard-Conway invariant to all two-component links.
problem Counting irreducible SU(2) representations for two-component links.
method Counting irreducible SU(2) representations with fixed meridional traces.
result Invariant equals symmetrized multivariable link signature for (2, 2n)-torus links.
The paper extends CF-moves to classify virtual links of any number of components.
problem Classifying virtual links using CF-moves.
method Extending CF-moves to classify virtual links of arbitrary number of components using the virtual linking number and invariants.
result Classification of 3-component even virtual links up to CF-moves.
The paper confirms a conjecture and extends arrow polynomial to twisted links.
problem Extending classical invariants to virtual and twisted links.
method Checkerboard framings, cut points, and normalized arrow polynomial.
result The normalized arrow polynomial is an invariant for twisted links.
New obstructions show links with vanishing Milnor invariants may not be concordant to homology boundary links.
problem Understanding links with vanishing Milnor invariants and their concordance properties.
method Developing new obstructions and examples within the solvable filtration framework.
result Existence of links with vanishing Milnor invariants that are not concordant to homology boundary links.
New examples show limits of physical link isotopies.
problem Limits of physical link isotopies in thick and thickly embedded links.
method Construction of specific examples of thick and thickly embedded links.
result Explicit examples of links that cannot be split through thick homotopies.
It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are give…
Researchers compute link surgery modules for 2-component L-space links.
problem Computing the link surgery modules of 2-component L-space links.
method Koszul duality to extend earlier computations.
result Computed entire link surgery modules modulo a technical result.
We show that if a split link is obtained from a split link L in S3 by 1/n-Dehn surgery along a trivial knot C, then the link L∪C is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial li…
Paper proves link diagrams can be realized for some but not all types of links.
problem Realizing link diagrams for different types of links.
method Analyzing link diagrams for welded and virtual links.
result Similar results hold for welded links but not for virtual links.
Free surface-links are shown to be ribbon links.
problem Characterizing surface-links as ribbon links.
method Four proofs are provided, including a stabilization approach.
result Every free surface-link is a ribbon surface-link.
Study links' arc index and Turaev genus, proving conjectures.
problem Understanding the arc index and Turaev genus of links.
method Computed arc index, established bounds, and conjectured inequalities.
result Proved conjectures linking crossing number, arc index, and Turaev genus.
Given an m-component link L in S3 (m≥2), we construct a family of links which are link homotopic, but not link isotopic, to L. Every proper sublink of such a link is link isotopic to the corresponding sublink of L. Moreover, if L is an unlink then there exist links that in addition to the above prope…
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…
A method to convert pretzel links into braids.
problem Representing pretzel links as braids.
method General procedure to convert any pretzel link into a braid.
result A specific braid word corresponding to the pretzel link.
We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's s-invariant. We also study almost positive links, in particular, determine the s-invariants of almost positive links. This result suggests that all almost positive links might be strongl…
Generative Link Sequence Modeling predicts future links in evolving networks.
problem Predicting future links in networks with evolving structures.
method Sequence modeling framework with self-tokenization to capture temporal link formation patterns.
result GLSM achieves best performance on AUC metrics compared to existing methods.