Links Of Singularities Research Articles

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Suppose C is a singular curve in \mathbb CP^2 and it is topologically an embedded surface of genus g ; such curves are called cuspidal. The singularities of C are cones on knots K_i . We apply Heegaard Floer theory to find new constraints on the sets of knots \{K_i\} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d > 33 , that possess exactly one singularity which has exactly one Puiseux pair (p;q) . The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci sequence.

Links of rational singularities, L-spaces and LO fundamental groups

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 result of Hanselman et al. (Taut foliations on graph manifolds, 2015. arXiv:1508.0591), proving the conjecture of Boyer et al. (Math Ann 356(4):1213–1245, 2013), shows that a singularity link is not rational if and only if its fundamental group is left-orderable if and only if it admits a coorientable taut foliation.

Cutting arcs for torus links and trees

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$) among all positive tree-like Hopf plumbings.

Delta-Matroids and Vassiliev Invariants

Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a chord diagram depends on the intersection graph of the diagram rather than on the diagram itself. Moreover, it was shown that the value of this weight system on an intersection graph depends on the cy- cle matroid of the graph rather than on the graph itself. This result arose the question whether there is a natural way to introduce a 4-term relation on the space spanned by matroids, similar to the one for graphs. It happened however that the answer is negative: there are graphs having isomorphic cycle matroids such that applying the Vassiliev move to a pair of corresponding vertices a;b of the graphs we obtain two graphs with nonisomorphic matroids. The goal of the present paper is to show that the situation is different for binary delta-matroids: one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce a 4-term relation for them in such a way that the mapping taking a chord diagram to its delta-matroid respects the corresponding 4-term relations. Moreover, this mapping admits a natural extension to chord diagrams on several circles, which correspond to singular links. Delta-matroids were introduced by A. Bouchet for the purpose of studying embedded graphs, whence their relationship with (multiloop) chord diagrams is by no means unexpected. Some evidence for the existence of such a relationship can be found, for example, the Tutte polynomial for embedded graphs has been introduced. It was shown that this polynomial depends on the delta-matroid of the embedded graph rather than the graph itself and satises the Vassilev 4-term relation.

Three approaches to a bracket polynomial for singular links

Three approaches to a bracket polynomial for singular links

Singular links and Yang-Baxter state models

We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Speci\-fically, we consider a Yang-Baxter state model for the {\rm 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 representation of the singular braid monoid into a matrix algebra and seek conditions for further extending the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the {\rm sl($n$)} polynomial for classical links.

Links of plane curve singularities are L–space links

We prove that a sufficiently large surgery on any algebraic link is an L-space. For torus links we give a complete classification of integer surgery coefficients providing L-spaces.

On the link invariants from the Yokonuma–Hecke algebras

In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

Virtual singular braids and 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 generators.

Movie moves for singular link cobordisms in 4-dimensional space

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 projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

Delta-Matroids and Vassiliev Invariants

Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a chord diagram depends on the intersection graph of the diagram rather than on the diagram itself. Moreover, it was shown that the value of this weight system on an intersection graph depends on the cy- cle matroid of the graph rather than on the graph itself. This result arose the question whether there is a natural way to introduce a 4-term relation on the space spanned by matroids, similar to the one for graphs. It happened however that the answer is negative: there are graphs having isomorphic cycle matroids such that applying the Vassiliev move to a pair of corresponding vertices a;b of the graphs we obtain two graphs with nonisomorphic matroids. The goal of the present paper is to show that the situation is different for binary delta-matroids: one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce a 4-term relation for them in such a way that the mapping taking a chord diagram to its delta-matroid respects the corresponding 4-term relations. Moreover, this mapping admits a natural extension to chord diagrams on several circles, which correspond to singular links. Delta-matroids were introduced by A. Bouchet for the purpose of studying embedded graphs, whence their relationship with (multiloop) chord diagrams is by no means unexpected. Some evidence for the existence of such a relationship can be found, for example, the Tutte polynomial for embedded graphs has been introduced. It was shown that this polynomial depends on the delta-matroid of the embedded graph rather than the graph itself and satises the Vassilev 4-term relation.

A escolha e o reconhecimento pela educação: o caso de Antônio

Resumo O artigo sistematiza ganhos teóricos e metodológicos do uso da noção híbridas disposições de habitus a partir de um estudo de caso. Para desenvolver o argumento, considera primeiramente que a escolha pelo projeto escolar, em função da conquista de um reconhecimento social, pode auxiliar na explanação do princípio hierárquico da construção de um híbrido sistema orientador de conduta. A análise permite observar que o biografado pouco a pouco traça sua singularidade fundindo disposições variadas, entre elas a submissão moral proveniente do trabalho braçal, bem como disposições intelectuais e cognitivas provenientes da escola. Injunções históricas e específicas chances de escolarização foram capturadas e apropriadas por ele num jogo e uso singulares do saber difuso disponibilizado pela frequência a vários círculos sociais. Apontando aspectos do comportamento em várias dimensões de sua vida, como práticas éticas, estéticas, religiosas e profissionais, é possível considerar uma articulação singular de disposições de cultura. Por fim, a reflexão permite identificar as relações indissociáveis entre contexto socializador e processos de individuação, na medida em que condições de socialização específicas são capazes de instrumentalizar a composição singular dos indivíduos.

The mechanism of the keyhole lesion reassessed: An experimental approach

The initial description of the keyhole defect was detailed as a peculiar gunshot entrance wound in the cranial vault due to firearm discharge in a tangential path. This injury may be described in two parts: a rounded section with inner table beveling and a triangular section with outer table beveling.We report a case of a gunshot skull wound “keyhole” shaped, appeared to have been made perpendicularly to the bone. Performing an experimental study on cranial bones with shots made perpendicularly to the skull approved this hypothesis, and bone injuries were then architecturally characterised using high-resolution micro computed tomography.The singular link between the tangential gunshot path and the keyhole pattern has been refuted several times, and some authors have hypothesised that there is an important role for concentric fractures that occur far away from the initial impact point of the bullet. Micro computed tomography analysis of the bone showed these keyhole defect features with a particular high description. Then, the whole pattern has a spider-web effect, and these concentric fractures could explain the keyhole pattern even in a perpendicular gunshot path.

Cutting arcs for torus links and trees

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., An , Dn , E6 , E7 and E8 ) among all positive tree-like Hopf plumbings.

On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

Abstract This is a survey article about the study of the links of some complex hypersurface singularities in ℂ3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

Open-book decompositions of 𝕊5 and real singularities

In this article, we study the topology of the family of real analytic germs F : (ℂ3, 0) → (ℂ, 0) given by [Formula: see text] with p, q, r ∈ ℕ, p, q, r ≥ 2 and (p, q) = 1. Such a germ has an isolated singularity at 0 and gives rise to a Milnor fibration [Formula: see text]. Moreover, it is known that the link LF is a Seifert manifold and that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of 𝕊5 given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ℂ3.

The Canonical Contact Structure on the Link of a Cusp Singularity

Caubel, Nemethi, and Popescu-Pampu in [2] proved that an oriented 3-manifold admits at most one positive contact structure which can be realized as the complex tangency along the link of a complex surface singularity. They call it the Milnor fillable contact structure. Lekili and Ozbagci in [10] showed that a Milnor fillable contact structure is universally tight. In particular, by Honda's classification [5], the link of a cusp singularity is contactomorphic to the positive contact structure associated to the Anosov flow on a Sol-manifold (see [1]). We describe the contact structure on the link of a cusp singularity in two different ways without using Honda's classification theorem. One description is based on the toric method introduced in Mori [15]. The other description is based on Hirzebruch's construction of the Hilbert modular cusps. Consequently, we give certain answers to the problems in Mori [14] concerning the relation between the cusp singularities and the simple elliptic singularities, and the higher dimensional extension of the local Lutz-Mori twist.

A new proof of Vassiliev's conjecture

In the paper [First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in ℝn, Izv. Math. 69(5) (2005) 865–912] on finite type invariants of self-intersecting curves, Vassiliev conjectured a criterion of planarity of framed four-valent graphs, i.e. 4-graphs with an opposite edge structure at each vertex. The conjecture was proved by Manturov [A proof of Vassilievs conjecture on the planarity of singular links, Izv. Math. 69(5) (2005) 169–178]. We give here another proof of Vassiliev's planarity criterion of framed four-valent graphs (and more generally, (even) *-graphs), which is based on Pontryagin–Kuratowski theorem.

Fundamental groups of links of isolated singularities

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G G there is a complex projective surface S S with simple normal crossing singularities only, so that the fundamental group of S S is isomorphic to G G . We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to G G . Lastly, we prove that a finitely-presented group G G is Q {\mathbb Q} -superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if G G is isomorphic to the fundamental group of the link of a rational 6-dimensional complex singularity.

Canonical contact structures on some singularity links

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 bundle.