Extending the Sentence Verification Technique to Tables and Node-Link Diagrams

Author(s):  
Mark A. Livingston ◽  
Derek Brock ◽  
Tucker Maney ◽  
Dennis Perzanowski
Keyword(s):  
Author(s):  
Heather M Russell ◽  
Julianna Tymoczko

Abstract Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.


1991 ◽  
Vol 142 (3) ◽  
pp. 643-643
Author(s):  
Paolo Cotta-Ramusino ◽  
Maurizio Rinaldi

2011 ◽  
Vol 15 (4) ◽  
pp. 549-563 ◽  
Author(s):  
Colin Adams ◽  
Reiko Shinjo ◽  
Kokoro Tanaka
Keyword(s):  

Author(s):  
Shalin Hai-Jew

On the surface spaces of the WWW and Internet, organizations and individuals have long created a public face to emphasize their respective brands, showcase their credibility, and interact with others in often very public ways. These surface spaces include Websites, social media platforms, virtual worlds, interactive game spaces, content sharing sites, social networking sites, microblogging sites, wikis, blogs (Web logs), collaborative work sites, and email systems. Beneath the glittering surfaces are electronic understructures, which enable the mapping of networks (based on physical location or organization or URL), the tracking of inter-personal relationships between various accounts, the geolocation of various electronic data to the analog physical world, the de-anonymizing of aliases (to disallow pseudonymity), and the tracking of people to their contact information (digital and physical). Maltego Radium is a penetration testing tool that enables such crawls of publicly available information or Open-Source Intelligence (OSINT) to identify and describe electronic network structures for a range of applications. Further, this information is represented in a number of interactive node-link diagrams in both 2D and 3D for further insights. There is also an export capability for full reportage of the extracted information. This chapter introduces the tool and identifies some practical ways this has been used to “package” fresh understandings for enhanced awareness and decision-making in a higher education context.


1993 ◽  
Vol 02 (02) ◽  
pp. 233-237 ◽  
Author(s):  
MASAAKI WADA
Keyword(s):  

A method of coding diagrams of knots, links and tangles is introduced. Also, how to draw a diagram for a given code is explained.


2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.


2019 ◽  
Vol 28 (11) ◽  
pp. 1940001 ◽  
Author(s):  
Colin Adams ◽  
Jim Hoste ◽  
Martin Palmer

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.


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