scholarly journals A COMBINATORIAL DGA FOR LEGENDRIAN KNOTS FROM GENERATING FAMILIES

2013 ◽  
Vol 15 (02) ◽  
pp. 1250059 ◽  
Author(s):  
MICHAEL B. HENRY ◽  
DAN RUTHERFORD
Keyword(s):  
Standard Form ◽  
Geometric Interpretation ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Complex Sequence ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
Definition Of ◽  
Morse Complex ◽  
Differential Counts

For a Legendrian knot L ⊂ ℝ3, with a chosen Morse complex sequence (MCS), we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family F, we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov–Eliashberg DGA after changing coordinates by an augmentation.

scholarly journals Computing homology invariants of Legendrian knots

2014 ◽  
Vol 23 (11) ◽  
pp. 1450056 ◽  
Author(s):  
Emily E. Casey ◽  
Michael B. Henry
Keyword(s):  
Homology Group ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Combinatorial Interpretation ◽  
Knot Invariants ◽  
Homology Groups ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
The Many ◽  
Morse Complex

The Chekanov–Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov–Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This paper gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence (MCS). First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| ≤ 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff [The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math.224(1) (2006) 141–150] need not have isomorphic homology groups.

scholarly journals Invariants of Legendrian Knots in Circle Bundles

2003 ◽  
Vol 05 (04) ◽  
pp. 569-627 ◽  
Author(s):  
Joshua M. Sabloff
Keyword(s):  
Contact Structure ◽  
Homology Class ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Circle Bundle ◽  
Differential Graded Algebra ◽  
Circle Bundles ◽  
Standard Contact ◽  
Definition Of

Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

scholarly journals INVARIANTS FOR LEGENDRIAN KNOTS IN LENS SPACES

2011 ◽  
Vol 13 (01) ◽  
pp. 91-121 ◽  
Author(s):  
JOAN E. LICATA
Keyword(s):  
Cyclic Group ◽  
Group Action ◽  
Riemann Surfaces ◽  
Lens Spaces ◽  
Graded Algebra ◽  
Cyclic Cover ◽  
Legendrian Knots ◽  
Differential Graded Algebra ◽  
Cyclic Group Action

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra [Formula: see text] which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S1-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

scholarly journals Haemostatic and thrombo-embolic complications in pregnant women with COVID-19: a systematic review and critical analysis

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Juliette Servante ◽  
Gill Swallow ◽  
Jim G. Thornton ◽  
Bethan Myers ◽  
Sandhya Munireddy ◽  
...  
Keyword(s):  
Venous Thrombosis ◽  
Pregnant Women ◽  
Clinical Evidence ◽  
Case Reports ◽  
Standard Form ◽  
Test Results ◽  
Maternal Deaths ◽  
High Clinical Suspicion ◽  
Increased Risk ◽  
Definition Of

Abstract Background As pregnancy is a physiological prothrombotic state, pregnant women may be at increased risk of developing coagulopathic and/or thromboembolic complications associated with COVID-19. Methods Two biomedical databases were searched between September 2019 and June 2020 for case reports and series of pregnant women with a diagnosis of COVID-19 based either on a positive swab or high clinical suspicion where no swab had been performed. Additional registry cases known to the authors were included. Steps were taken to minimise duplicate patients. Information on coagulopathy based on abnormal coagulation test results or clinical evidence of disseminated intravascular coagulation (DIC), and on arterial or venous thrombosis, were extracted using a standard form. If available, detailed laboratory results and information on maternal outcomes were analysed. Results One thousand sixty-three women met the inclusion criteria, of which three (0.28, 95% CI 0.0 to 0.6) had arterial and/or venous thrombosis, seven (0.66, 95% CI 0.17 to 1.1) had DIC, and a further three (0.28, 95% CI 0.0 to 0.6) had coagulopathy without meeting the definition of DIC. Five hundred and thirty-seven women (56%) had been reported as having given birth and 426 (40%) as having an ongoing pregnancy. There were 17 (1.6, 95% CI 0.85 to 2.3) maternal deaths in which DIC was reported as a factor in two. Conclusions Our data suggests that coagulopathy and thromboembolism are both increased in pregnancies affected by COVID-19. Detection of the former may be useful in the identification of women at risk of deterioration.

scholarly journals Point configurations, phylogenetic trees, and dissimilarity vectors

2021 ◽  
Vol 118 (12) ◽  
pp. e2021244118
Author(s):  
Alessio Caminata ◽  
Noah Giansiracusa ◽  
Han-Bom Moon ◽  
Luca Schaffler
Keyword(s):  
Phylogenetic Trees ◽  
Geometric Interpretation ◽  
Explicit Characterization ◽  
Point Configurations ◽  
Definition Of ◽  
Tropical Basis ◽  
The Way

In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

scholarly journals GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

scholarly journals Teorier om dufortællinger: En blindgyde?

2011 ◽  
Vol 39 (112) ◽  
pp. 107-132
Author(s):  
Rolf Reitan
Keyword(s):  
Standard Form ◽  
Second Person ◽  
La Modification ◽  
Definition Of ◽  
Strict Definition

THEORIZING SECOND-PERSON NARRATIVES: A BACKWATER PROJECT? | In this paper Rolf Reitan proposes a closer look at three very different perspectives on second person narrative: Brian Richardson, Irene Kacandes, and Monika Fludernik have been classical references for some time, but they have never, according to Reitan, been seriously discussed. The paper begins by examining Kacandes’ intriguing concept of ‘radical narrative apostrophe’, and then discusses the three authors’ very different typological proposals. Borrowing Richardson’s idea of a Standard Form of second person narration, it returns to Butor’s La Modification to investigate the question of address (a pivotal question in Fludernik’s articles), which then leads to a strict definition of a prototypical “genre” of Standard Form narratives. Passing through conceptual landscapes of fiction, apostrophe, and postmodernism, some tricky questions concerning selfaddress,and some of Margolin’s analytic formulas, are considered. At last, by way of proposing a much needed subdivision of the Standard Form, Reitan discusses the strange narrating voice in La modification: not a narratorial voice, but a readerly voice created in the author’s writing.

Whose Standard, Which Model? Towards the Definition of a Standard Nigerian Spoken English for Teaching, Learning and Testing in Nigerian Schools

1990 ◽  
Vol 89-90 ◽  
pp. 91-106
Author(s):  
V.O. Awonusi
Keyword(s):  
Native Speaker ◽  
Standard Form ◽  
Teaching English ◽  
Spoken English ◽  
Social Variation ◽  
Local Standard ◽  
English Pronunciation ◽  
Definition Of ◽  
Teaching Learning

Abstract The adoption of RP as a model of teaching in non-native speaker English societies such as Nigeria seems to have come to say. However, the accent of English that emerged in Nigeria, over the years, (to some linguists) is anything but RP (although some hold the view that there are a few RP speakers in Nigeria). We are, therefore, forced to ask the question: What is RP?; or better still: what are its defining characteristics, particularly in relation to non-native varieties of English? Consequently, were are motivated to search for, and identify alternative local (standard) accents for teaching purposes. This paper attempts to identify the Nigerian standard accent of English that is appropriate for adoption as a model for teaching English pronunciation. It examines the problems associated with the identification of a standard form in the light of variables such as international intelligibility local acceptability, a real and social variation, in native and non-native speaker communities alike. On the basis of socio-linguistic realities it recommends an accent for teaching, learning and testing in Nigerian schools.

Sign in / Sign up

Export Citation Format

Share Document