scholarly journals Classification of Legendrian knots of topological type 76 with maximal Thurston–Bennequin number

2019 ◽  
Vol 28 (14) ◽  
pp. 1950089
Author(s):  
Ivan Dynnikov ◽  
Maxim Prasolov
Keyword(s):  
Topological Type ◽  
Legendrian Knots ◽  
Type Formula

We classify Legendrian knots of topological type [Formula: see text] having maximal Thurston–Bennequin number confirming the corresponding conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997].

scholarly journals The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros

2016 ◽  
Vol 60 (2) ◽  
pp. 319-348 ◽  
Author(s):  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa ◽  
Juan J. Nuño-Ballesteros
Keyword(s):  
Structure Theorem ◽  
Topological Type ◽  
Topological Invariant ◽  
Topological Equivalence ◽  
Stable Maps ◽  
Reeb Graph ◽  
Cone Structure ◽  
Topological Description ◽  
Stable Map

AbstractWe consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

Deformed mesh algebras of Dynkin type 𝔽4

2019 ◽  
Vol 19 (03) ◽  
pp. 2050049
Author(s):  
Jerzy Białkowski
Keyword(s):  
Type Formula ◽  
Dynkin Type ◽  
Mesh Algebra

This paper belongs to the series of articles devoted to the classification of deformed mesh algebras of Dynkin types [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We prove here that every deformed mesh algebra of type [Formula: see text] is isomorphic to the canonical mesh algebra of type [Formula: see text].

scholarly journals GLOBAL CLASSIFICATION OF A CLASS OF CUBIC VECTOR FIELDS WHOSE CANONICAL REGIONS ARE PERIOD ANNULI

2011 ◽  
Vol 21 (07) ◽  
pp. 1831-1867 ◽  
Author(s):  
M. CAUBERGH ◽  
J. LLIBRE ◽  
J. TORREGROSA
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Vector Fields ◽  
Topological Type ◽  
Radial Symmetry ◽  
Phase Portraits ◽  
Arbitrary Degree ◽  
Systematic Classification ◽  
Global Classification

We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincaré disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

scholarly journals Real Algebraic Curves on Real del Pezzo Surfaces

2020 ◽  
Author(s):  
Matilde Manzaroli
Keyword(s):  
Algebraic Curves ◽  
Topological Type ◽  
Del Pezzo Surfaces ◽  
Toric Surfaces ◽  
Real Algebraic Curves ◽  
Del Pezzo ◽  
Classical Subject ◽  
The 19Th Century ◽  
Type Classification

Abstract The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein, and Hilbert in the 19th century; in particular, the isotopy-type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is known for more general ambient surfaces. We take a step forward in the study of topological-type classification of real algebraic curves on non-toric surfaces focusing on real del Pezzo surfaces of degree 1 and 2 with multi-components real part. We use degeneration methods and real enumerative geometry in combination with variations of classical methods to give obstructions to the existence of topological-type classes realized by real algebraic curves and to give constructions of real algebraic curves with prescribed topology.

Foliated compressing discs for Legendrian rational tangles

2016 ◽  
Vol 25 (06) ◽  
pp. 1650029
Author(s):  
Gregory R. Schneider
Keyword(s):  
Contact Structure ◽  
Legendrian Knots ◽  
Ongoing Effort ◽  
Rational Tangles ◽  
Standard Contact ◽  
New Framework

We establish a new framework for diagramming both Legendrian rational tangles in the standard contact structure on [Formula: see text] and the signed characteristic foliations of their associated compressing discs, as well as the technical means by which these diagrams can be used to study Legendrian isotopies of such tangles. We then establish a number of results that represent new progress in the ongoing effort to classify Legendrian rational tangles under a pair of operations known as Legendrian flypes. These operations, while topologically isotopies, are known to produce distinct Legendrian objects in many circumstances, a fact that has been of much interest throughout the study and classification of Legendrian knots.

scholarly journals Generalized null correlation bundles

1988 ◽  
Vol 111 ◽  
pp. 13-24 ◽  
Author(s):  
Lawrence Ein
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Topological Type ◽  
Stable Rank ◽  
Chern Classes ◽  
Smooth Variety ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

It is well known that the moduli space of stable rank 2 vector bundles on ℙ2 of the fixed topological type is an irreducible smooth variety ([1], and [8]). There are also many known results on the classification of stable rank 2 vector bundles on ℙ3 with “small” Chern classes.

scholarly journals Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

2018 ◽  
Vol 17 (12) ◽  
pp. 1850237
Author(s):  
Seth Shelley-Abrahamson ◽  
Alec Sun
Keyword(s):  
Irreducible Representations ◽  
Infinite Dimensional ◽  
Cherednik Algebras ◽  
Type Formula ◽  
Type D ◽  
Finite Dimensional ◽  
Rational Cherednik Algebra ◽  
Wall Crossing ◽  
Rational Cherednik Algebras

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations [Formula: see text] of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] for symmetric bipartitions [Formula: see text] are infinite dimensional for all parameters [Formula: see text]. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text] arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text].

Note on the spectral classification of carbon stars in the photographic infrared region

1966 ◽  
Vol 24 ◽  
pp. 21-23
Author(s):  
Y. Fujita
Keyword(s):  
Infrared Region ◽  
Spectral Classification ◽  
Carbon Stars

We have investigated the spectrograms (dispersion: 8Å/mm) in the photographic infrared region fromλ7500 toλ9000 of some carbon stars obtained by the coudé spectrograph of the 74-inch reflector attached to the Okayama Astrophysical Observatory. The names of the stars investigated are listed in Table 1.

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