Links Of Singularities Research Articles

Links of Singularities of Inner Non-degenerate Mixed Functions

Links of Singularities of Inner Non-degenerate Mixed Functions

On fillings of contact links of quotient singularities

AbstractWe study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co‐fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non‐existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including for , which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of , and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

A monopole invariant for families of contact structures

We use monopole Floer homology to study the topology of the space of contact structures on a 3-manifold. Our main tool is a generalisation of the Kronheimer–Mrowka–Ozsváth–Szabó contact invariant to an invariant for families of contact structures, and we establish foundational results that describe the interaction between this invariant and the module structure in monopole Floer homology. We apply these results in several examples of contact manifolds, such as the links of non-rational surface singularities, to deduce several applications. Namely, we are able to obstruct the existence of sections of a natural fibration over the 2-sphere whose total space is the space of contact structures on the 3-manifold, and from this we are able to detect the existence of exotic loops of contact structures on contact 3-manifolds with convex sphere boundary.

Shifts in population density centers of a hibernating mammal driven by conflicting effects of climate change and disease.

Populations wax and wane over time in response to an organism's interactions with abiotic and biotic forces. Numerous studies demonstrate that fluctuations in local populations can lead to shifts in relative population densities across the geographic range of a species over time. Fewer studies attempt to disentangle the causes of such shifts. Over four decades (1983-2022), we monitored populations of hibernating Indiana bats (Myotis sodalis) in two areas separated by ~110 km. The number of bats hibernating in the northern area increased from 1983 to 2011, while populations in the southern area remained relatively constant. We used simulation models and long-term weather data to demonstrate the duration of time bats must rely on stored fat during hibernation has decreased in both areas over that period, but at a faster rate in the northern area. Likewise, increasing autumn and spring temperatures shortened the periods of sporadic prey (flying insect) availability at the beginning and end of hibernation. Climate change thus increased the viability of northern hibernacula for an increasing number of bats by decreasing energetic costs of hibernation. Then in 2011, white-nose syndrome (WNS), a disease of hibernating bats that increases energetic costs of hibernation, was detected in the area. From 2011 to 2022, the population rapidly decreased in the northern area and increased in the southern area, completely reversing the northerly shift in population densities associated with climate change. Energy balance during hibernation is the singular link explaining the northerly shift under a changing climate and the southerly shift in response to a novel disease. Continued population persistence suggests that bats may mitigate many impacts of WNS by hibernating farther south, where insects are available longer each year.

Disingquandles and invariants of dichromatic singular links

We introduce and investigate dichromatic singular links. We also construct disingquandles and use them to define counting invariants for unoriented dichromatic singular links. We provide some examples to show that these invariants distinguish some dichromatic singular links.

What can be done about solitude? From injunction to isolation, the production of a singular link

What can be done about solitude? From injunction to isolation, the production of a singular link

Pseudo links and singular links in the Solid Torus

In this paper we introduce and study the theories of pseudo links and singular links in the Solid Torus, ST. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams, and that have potential use in molecular biology, while singular links are links that contain a finite number of self-intersections. We consider pseudo links and singular links in ST and we set up the appropriate topological theory in order to construct invariants for these types of links in ST. In particular, we formulate and prove the analogue of the Alexander theorem for pseudo links and for singular links in ST. We then introduce the mixed pseudo braid monoid and the mixed singular braid monoid, with the use of which, we formulate and prove the analogue of the Markov theorem for pseudo links and for singular links in ST. \smallbreak Moreover, we introduce the pseudo Hecke algebra of type A, $P\mathcal{H}_n$, the cyclotomic and generalized pseudo Hecke algebras of type B, $P\mathcal{H}_{1, n}$, and discuss how the pseudo braid monoid (cor. the mixed pseudo braid monoid) can be represented by $P\mathcal{H}_{n}$ (cor. by $P\mathcal{H}_{1, n}$). This is the first step toward the construction of HOMFLYPT-type invariants for pseudo links in $S^3$ and in ST. We also introduce the cyclotomic and generalized singular Hecke algebras of type B, $S\mathcal{H}_{1, n}$, and we present two sets that we conjecture that they form linear bases for $S\mathcal{H}_{1, n}$. Finally, we generalize the bracket polynomial for pseudo links in ST.

On the axioms of singquandles

In this paper, we deal with the notion of singquandles introduced in [I. R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications 26(14) (2017) 1750099]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandles. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case).

On uniqueness of symplectic fillings of links of some surface singularities

On uniqueness of symplectic fillings of links of some surface singularities

Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.

Refined disk potentials for immersed Lagrangian surfaces

We define a refined Gromov–Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4‑manifold that are self-transverse as an element in a capped version of the Chekanov–Eliashberg dg‑algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5‑sphere.

Homflypt skein theory, string topology and 2-categories

We show that relations in Homflypt type skein theory of an oriented [Formula: see text]-manifold [Formula: see text] are induced from a [Formula: see text]-groupoid defined from the fundamental [Formula: see text]-groupoid of a space of singular links in [Formula: see text]. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental [Formula: see text]-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.

Singquandle shadows and singular knot invariants

AbstractWe introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.

An Estimation and Evaluation of Network Availability in Link State Routing Networks

The internet is taking component in a developing feature in every non-public and professional activity. The real-time, delay sensitive and mission-essential purposes, community availability requirement is beforehand for internet carrier providers (ICPs). The loop-loose criterion (LLC) approach has been extensively deployed through numerous ICPs for handling the best network component failure state of affairs in fantastic internet through. The achievement of LLC lies in its inherent simplicity; however, this comes at the rate of letting certain failure. To reap complete failure safety with LLC without incurring significant extra, a singular link protection scheme, hybrid hyperlink protocol (HLP), to reap failure routing. In contrast with in advance schemes, HLP guarantees tall network in a greater surroundings pleasant way. HLP is carried out in stages. Initial level substances a surroundings pleasant LLC primarily based totally on (MNP-e). The complexity of the set of rules is decrease than that of Dijkstra’s set of rules and might gift similar to network availability with LCC (Loop-loose criterion). Moment level substances backup direction safety based on MNP-e, the area totally a minimum type of need to be protected, to fulfill the network requirement. We don't forget those algorithms in a massive spread of associated, real and actual, and the effects display that HLP can achieve lofty network without introducing apparent.

Tied Pseudo Links & Pseudo Knotoids

In this paper, we study the theory of pseudo knots, which are knots with some missing crossing information, and we introduce and study the theory of pseudo tied links and the theory of pseudo knotoids. In particular, we first present a braiding algorithm for pseudo knots and we then introduce the L-moves in that setting, with the use of which we formulate a sharpened version of the analogue of the Markov theorem for pseudo braids. Then, we introduce and study the theory of tied pseudo links, that generalize the notion of tied links, and we exploit the relation between tied pseudo links and tied singular links. We first present an L-move braid equivalence for tied singular braids. Then, we introduce the tied pseudo braid monoid and we formulate and prove analogues of the Alexander and Markov theorems for tied pseudo links. Finally, we introduce and study the theory of pseudo knotoids, that generalize the notion of knotoids. We present an isotopy theorem for pseudo knotoids and we then pass to the level of braidoids. We further introduce and study the pseudo braidoids by introducing the pseudo L-moves and by presenting the analogues of the Alexander and Markov theorems for pseudo knotoids. We also discuss further research related to tied (multi)-knotoids and tied pseudo (multi)-knotoids. The theory of pseudo knots may serve as a strong tool in the study of DNA, while tied links have potential use in other aspects of molecular biology.

Tied monoids

We construct certain monoids, called tied monoids. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids indicate that they should give origin to new knot algebras; indeed, our tied monoids include the tied braid monoid and the tied singular braid monoid, which were used, respectively, to construct new polynomial invariants for classical links and singular links. Consequently, we provide a mechanism to attach an algebra to each tied monoid; this mechanism not only captures known generalizations of the bt-algebra, but also produces possible new knot algebras. To build the tied monoids it is necessary to have presentations of set partition monoids of types A, B and D, among others. For type A we use a presentation due to FitzGerald and for the other type it was necessary to built them.

SYMPLECTIC FILLINGS OF QUOTIENT SURFACE SINGULARITIES AND MINIMAL MODEL PROGRAM

We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic varieties.

Surface singularities and planar contact structures

Dans cet article on démontre que si une structure de contact sur une variété de dimension trois est portée par un livre ouvert à pages planaires, alors une certaine configuration d’intersections n’apparaît dans l’homologie d’aucun de ses remplissages minimaux. On démontre de plus que les remplissages d’une telle variété de contact ne contiennent pas de surface symplectique de genre positif. En appliquant ces obstructions aux structures de contact canoniques sur les bords des singularités normales de surfaces, on montre que les bords des singularités isolées de surfaces dans l’espace complexe de dimension trois sont planaires seulement pour les singularités de type A n . En général, nous caractérisons complètement les bords planaires des singularités normales de surfaces (par leurs graphes de résolution) : ces singularités sont précisément les singularités rationnelles avec cycle fondamental réduit (aussi appelées singularités minimales). On montre aussi la non-planarité des structures de contact tendues sur certains petits L-espaces de Seifert ainsi que celle des structures de contact obtenues par la construction de Boothby–Wang appliquée aux surfaces de genre positif. De plus, on démontre que tout groupe de présentation finie est le groupe fondamental d’une fibration de Lefschetz à fibres planaires.

Heegaard Floer invariants of contact structures on links of surface singularities

Let a contact 3-manifold (Y, \xi_0) be the link of a normal surface singularity equipped with its canonical contact structure \xi_0 . We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant c^+(\xi_0)\in \mathrm {HF}^+(-Y) cannot lie in the image of the U -action on \mathrm {HF}^+(-Y) . It follows that Karakurt's "height of U -tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of U -tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and Némethi's lattice cohomology.

Tied links and invariants for singular links

Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic–combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi–direct group product. By using this decomposition we reprove the Alexander and Markov theorem for tied links; also, we introduce the tied singular knots, the tied singular braid monoid and certain families of Homflypt type invariants for tied singular links; these invariants are five–variables polynomials. Finally, we study the behavior of these invariants; in particular, we show that our invariants distinguish non isotopic singular links indistinguishable by the Paris–Rabenda invariant.