arXiv research

Defines singular grid diagrams for various types of links.

Study on oriented disingquandles for distinguishing singular links.

New singquandles help distinguish certain types of links.

New polynomial invariant distinguishes singular links.

New algebraic invariants for singular links discovered.

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …

Generates moves for oriented singular links using algebraic structures.

New invariant for singular links via bt-algebra.

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from …

A generalised Thurston-Bennequin invariant for a Q-singularity of a real algebraic variety is defined as a linking form on the homologies of the real link of the singularity. The main goal of this paper is to present a method to calculate the linking form in terms of the very good resolution graph of a real normal unib…

Study on singular twisted links and virtual braids, extending knot theory concepts.

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

Study links between singular points and fibers of maps.

Invariants for singular knots and links using non-commutative cocycles.

New mixed singularities help classify real algebraic links.

Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.

Extended polynomial for virtual singular links, decomposing into two invariants.

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

New invariant distinguishes singular knots and links.

We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler c…

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

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…

The paper defines subgroups of camomile type and studies singular braids and links.

New method detects essential tori in mixed singularity links.

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…

Paper describes links of mixed polynomials with specific properties.

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We …

We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibilit…

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local rela…

Constructs Lagrangian skeleta for curve singularities.

Generalizes biquandles to psyquandles for singular and pseudolinks invariants.

Study on singularities of specific polynomial functions.

Divides help construct fibered links from singularities.

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$…

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

Study Stein and Milnor fillings of links from surface singularities.

We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a rep…

We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bu…

Aicardi's invariant $F(L)$ is extended to colored singular links using graphical calculus.

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…

We prove that the link of a complex normal surface singularity is an L--space if and only if the singularity is rational. This via a recent result of Hanselman, J. Rasmussen, S. D. Rasmussen and Watson (proving the conjecture of Boyer, Gordon and Watson), shows that a singularity link is not rational if and only if its…

Study shows infinite distinct outer metric Lipschitz classes for knots in $S^3$.

The paper proves a link concordance implies homotopy theorem in high codimensions.

New invariants for singular knots and links defined using shadow structures.

The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.

Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano…