scholarly journals The Legendrian knot complement problem

2018 ◽  
Vol 27 (14) ◽  
pp. 1850067 ◽  
Author(s):  
Marc Kegel
Keyword(s):  
Contact Structure ◽  
Legendrian Knot ◽  
Tight Contact ◽  
Legendrian Links ◽  
User Friendly ◽  
The Way

We prove that every Legendrian knot in the tight contact structure of the [Formula: see text]-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight [Formula: see text]-sphere. On the way a new user-friendly formula for computing the Thurston–Bennequin invariant of a Legendrian knot in a surgery diagram is given.

Geometry of Legendrian knots

2016 ◽  
Vol 25 (13) ◽  
pp. 1650069
Author(s):  
Dishant M. Pancholi ◽  
Suhas Pandit
Keyword(s):  
Contact Structure ◽  
Total Curvature ◽  
Legendrian Knots ◽  
Linking Number ◽  
Legendrian Knot ◽  
Tight Contact ◽  
Torus Knot ◽  
Extrinsic Geometry ◽  
Bounded Below ◽  
Explicit Relation

We study the extrinsic geometry of Legendrian knots in the standard tight contact structure on [Formula: see text] In particular, we show that the total curvature of a Legendrian knot [Formula: see text] in [Formula: see text] is bounded below by [Formula: see text] times, the total number of cusps in the front projection of [Formula: see text]. We also show that a Legendrian [Formula: see text]-torus knot has the total curvature bounded below by [Formula: see text] while that of the Legendrian knots [Formula: see text] is bounded below by [Formula: see text]. Furthermore, we find an explicit relation between the Thurston–Bennequin number of a Legendrian knot [Formula: see text] and the geometric self-linking number, the curvature and the torsion of the knot [Formula: see text].

scholarly journals ON THE S1 × S2 HOMFLY-PT INVARIANT AND LEGENDRIAN LINKS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350040 ◽  
Author(s):  
MIKHAIL LAVROV ◽  
DAN RUTHERFORD
Keyword(s):  
Rational Function ◽  
Contact Structure ◽  
Solid Torus ◽  
Laurent Polynomial ◽  
Skein Modules ◽  
Skein Module ◽  
Tight Contact ◽  
Legendrian Links ◽  
Simple Basis

In [On the HOMFLY-PT skein module of S1 × S2, Math. Z. 237(4) (2001) 769–814], Gilmer and Zhong established the existence of an invariant for links in S1 × S2 which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in S1 × S2 and shows that the invariant is in fact a Laurent polynomial in a and z = s – s-1. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston–Bennequin number to Legendrian links in S1 × S2 with its tight contact structure.

The Merging of Computers and Business Communication

1987 ◽  
Vol 15 (4) ◽  
pp. 383-389 ◽  
Author(s):  
Joan C. Roderick ◽  
Karen A. Forcht
Keyword(s):  
College Students ◽  
Organizational Communication ◽  
Teaching Strategies ◽  
Educational Experience ◽  
Business Communication ◽  
Computer Applications ◽  
Hard Copy ◽  
Hands On ◽  
User Friendly ◽  
The Way

Because of the availability of user-friendly software and the affordability of hardware, computers have become a common means of organizational communication. Users have had to change the way they process thoughts and ideas and to transfer them into hard-copy documentation. The integration of the computer into the business communication curriculum allows the instructor to provide a relevant and practical educational experience for college students. This article examines the importance of incorporating hands-on usage of a microcomputer in the business communication class and discusses computer applications and teaching strategies for text editing, punctuation review, and grammar assistance.

ELIV 2021

2021 ◽  
Keyword(s):  
Automotive Industry ◽  
Digital Ecosystems ◽  
Start Up ◽  
The Future ◽  
User Friendly ◽  
Full Swing ◽  
Level 2 ◽  
The Way

Foreword Start-up future It has felt like Covid-19 had a stranglehold on us. But we haven‘t allowed ourselves to be defeated. On the contrary, we are taking advantage of the opportunities that arise as a result. Not only the long-overdue push towards digitalization, for example, but also the time gained by making fewer journeys. Those who show strength now and position themselves for the future will win. And that is exactly the reason why we have been preparing ELIV 2021 with such a lot of enthusiasm. As usual, we have prepared an up-to-date program with the familiar mixture of technically demanding and strategic papers and are sure that the ELIV platform will once again be a trendsetter for the automotive industry. The CASE megatrends (Connected, Autonomous, Shared, Electric) continue to disrupt the industry. In the Connect environment, there is still a struggle for user-friendly services and competition amongst digital ecosystems is in full swing. The entry of powerful central computers into electronic architectures poses major challenges for all parties involved. On the way from Level 2 to Levels 3, 4 and 5 all manufacturers are cur...

scholarly journals Fiberwise convexity of Hill’s lunar problem

2017 ◽  
Vol 09 (04) ◽  
pp. 571-630 ◽  
Author(s):  
Junyoung Lee
Keyword(s):  
Energy Level ◽  
Contact Structure ◽  
Finsler Geometry ◽  
Critical Energy ◽  
Critical Value ◽  
Finsler Metrics ◽  
Tight Contact ◽  
Systolic Inequality ◽  
Geodesic Problem ◽  
Hill's Lunar Problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

Dynamical Mathematical Software

2013 ◽  
pp. 70-88
Author(s):  
Samer Habre
Keyword(s):  
Learning Environment ◽  
Difference Equations ◽  
Mathematical Software ◽  
Mathematical Concepts ◽  
Visual Component ◽  
Innovative Methods ◽  
Modern Learning ◽  
Teaching Of Mathematics ◽  
User Friendly ◽  
The Way

Understanding mathematical concepts is many-folded. Traditional mathematics mostly emphasizes the algebraic/analytical aspect of a problem with minimal reference to its graphical aspect and/or numerical one. In a modern learning environment, however, multiple representations of concepts are proving to be essential for the teaching of mathematics. The availability of user-friendly dynamical software programs has paved the way for a radical yet smooth way for changing the way mathematical concepts are perceived. This chapter presents some of the author’s attempts for employing innovative methods in teaching topics in calculus, in differential and difference equations. The focus is on the use of dynamical programs that boost the visual component of the topics being investigated, hence contributing to a more complete understanding of these topics.

Semantic Web Adaptation

2011 ◽  
pp. 78-88
Author(s):  
Alexander Mikroyannidis ◽  
Babis Theodoulidis
Keyword(s):  
Semantic Web ◽  
Web Sites ◽  
Web Usage Mining ◽  
New Era ◽  
Automatic Reasoning ◽  
Web Usage ◽  
User Friendly ◽  
Unstructured Information ◽  
The Web ◽  
The Way

The rate of growth in the amount of information available in the World Wide Web has not been followed by similar advances in the way this information is organized and exploited. Web adaptation seeks to address this issue by transforming the topology of a Web site to help users in their browsing tasks. In this sense, Web usage mining techniques have been employed for years to study how the Web is used in order to make Web sites more user-friendly. The Semantic Web is an ambitious initiative aiming to transform the Web to a well-organized source of information. In particular, apart from the unstructured information of today’s Web, the Semantic Web will contain machine-processable metadata organized in ontologies. This will enhance the way we search the Web and can even allow for automatic reasoning on Web data with the use of software agents. Semantic Web adaptation brings traditional Web adaptation techniques into the new era of the Semantic Web. The idea is to enable the Semantic Web to be constantly aligned to the users’ preferences. In order to achieve this, Web usage mining and text mining methodologies are employed for the semi-automatic construction and evolution of Web ontologies. This usage-driven evolution of Web ontologies, in parallel with Web topologies evolution, can bring the Semantic Web closer to the users’ expectations.

scholarly journals A Model for Using DataCite DOIs in Observatory Bibliographies

2018 ◽  
Vol 186 ◽  
pp. 12011 ◽  
Author(s):  
Arnold Rots ◽  
Raffaele D’ Abrusco ◽  
Sherry Winkelman
Keyword(s):  
Careful Consideration ◽  
User Friendly ◽  
The Way

The use of DOI persistent identifiers has become an attractive mechanism for citing datasets in articles. However, taking the users’ interests into account requires careful consideration of the way in which we apply these identifiers. Our objective is the application of DOIs in such a way that individual datasets are properly cited while presenting the citations to the reader in a user-friendly manner. This is achieved by making judicious use of the metadata structure provided by DataCite.

De verleidingen en gevaren van GrETEL*

2020 ◽  
Vol 25 (1) ◽  
pp. 7-37 ◽  
Author(s):  
Jan Odijk
Keyword(s):  
Query Language ◽  
Exact Nature ◽  
Precise Nature ◽  
Linguistic Research ◽  
User Friendly ◽  
Annotated Corpora ◽  
The Way

Abstract Corpora are a useful and important source of evidence for linguistic research, but they are not the only kind of evidence, do not have any special status as evidence, and have their limitations. Recent very user-friendly applications such as GrETEL make it very easy to search in large and richly annotated corpora on the basis of an example sentence and without knowledge of a query language or the exact nature of the linguistic annotations. It is therefore very tempting to use these applications intensively. That is fine, but also dangerous in ways, because in many cases, in order to interpret the results correctly, the researcher must really be aware of the precise nature of the linguistic annotations and of the way in which the user-friendly interface generates a query on the basis of an example sentence. I will illustrate this with several examples. I also sketch some methods for avoiding or mitigating the dangers and argue that the applications should support these methods also in as user-friendly a manner as possible.

scholarly journals Legendrian torus knots in S1 × S2

2015 ◽  
Vol 24 (12) ◽  
pp. 1550064 ◽  
Author(s):  
Feifei Chen ◽  
Fan Ding ◽  
Youlin Li
Keyword(s):  
Contact Structure ◽  
Torus Knots ◽  
Tight Contact

We classify Legendrian torus knots in S1 × S2 with its standard tight contact structure up to Legendrian isotopy.

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