arXiv research

This paper determines the braid indices for non-alternating pretzel links.

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite t…

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invar…

Calculates lower bounds for type III Reidemeister moves in link diagrams.

Study completes braid index determination for all pretzel links.

Finite type invariants separate PL links in 3D space.

The paper characterizes Conway-Coxeter friezes using rational links.

Defines homotopy type for links in thickened surfaces.

This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components.

Study finds many Lagrangian fillings for certain Legendrian links.

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the p…

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of…

In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …

New heuristics for predicting links in multiplex networks.

In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding spaces of ``chord diagrams''.

The paper extends knot polynomials to annular and toroidal pseudo links.

Polynomial bound on Reidemeister moves for each link type.

New homotopy types defined for links in thickened surfaces with higher genus.

The paper proves there are many Lagrangian fillings for Legendrian links of affine type.

The study finds many Lagrangian fillings for Legendrian links of specific types.

Paper constructs a HOMFLYPT-type invariant for pseudo links.

Paper explores conditions for constructing new quasi-alternating links.

New findings on T-links derived from torus links.

We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful nontrivial example of the second Steenrod square operator on the Khovanov homology for …

Defines a new link invariant for type D webs.

Study of generalized knots and links, proving inequality involving crossing number and braid index.

The paper studies pseudo and singular links in a solid torus, developing invariants and algebraic structures.

The paper explains why a specific type of link homology is useful.

This paper aims at the problem of link pattern prediction in collections of objects connected by multiple relation types, where each type may play a distinct role. While common link analysis models are limited to single-type link prediction, we attempt here to capture the correlations among different relation types and…

Computes knot types using HOMFLY-PT polynomial.

We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable …

Let $M_λ$ be the $λ$-component Milnor link. For $λ\ge 3$, we determine completely when a finite slope surgery along $M_λ$ yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite slope surgery} implies that a surgery coefficient of every component is not $\infty$. For $λ=3$ (i.e.\ the Borromean rings)…

We find two new classes of virtually fibered classic Montesinos links of type $\widetilde{SL_2}$.

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low de…

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

We use an action, of 2l-component string links on l-component string links, defined by the first author and Xiao-Song Lin, to lift the indeterminacy of finite type link invariants. The set of links up to this new indeterminacy is in bijection with the orbit space of the restriction of this action to the stabilizer of t…

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities…

Khovanov homology extended to 3-manifolds, linking tangles.

New algebraic structure for 2-string links and long knots.

Knot lattice homology invariant of smooth knot type in rational homology spheres.

Entity linking is the task of linking mentions of named entities in natural language text, to entities in a curated knowledge-base. This is of significant importance in the biomedical domain, where it could be used to semantically annotate a large volume of clinical records and biomedical literature, to standardized co…

We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric com…

Study on homology groups of cDV singularity links, identifying their topology.

We find a larger class of virtually fibred classic Montesinos links of type $\widetilde{SL_2}$, extending a result of Agol, Boyer and Zhang.

We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We…

New examples of hyperbolic links with generalized torsion elements found.

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, the first author proved that the restriction to Brunnian links of any Goussarov-Vassiliev finite type invariant of (n+1)-component links of degree<2n is trivial. The purpose of this paper is to study the first nont…

We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored…