COVERINGS OF LINKS AND A GENERALIZATION OF RILEY’S CONJECTURE B

Let k be a knot in S3, ψ: π1(S3–k)→Zα1 * Zα2 an epimorphism sending a meridian to an element of length 2 and ω: Zα1 * Zα2→Sσ a transitive representation into a symmetric group. Information is given (in Theorem B##) on the homologies of the unbranched and branched covers associated to ωψ which generalizes a conjecture of Riley. When k is a torus knot these homologies are actually computed.

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Representations of the Infinite Symmetric Group

10.1017/cbo9781316798577 ◽

2016 ◽

Cited By ~ 3

Author(s):

Alexei Borodin ◽

Grigori Olshanski

Keyword(s):

Symmetric Group ◽

Infinite Symmetric Group

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Processing of Social Identity Threats

Social Psychology ◽

10.1027/1864-9335/a000133 ◽

2013 ◽

Vol 44 (6) ◽

pp. 361-372 ◽

Cited By ~ 23

Author(s):

Natascha de Hoog

Keyword(s):

Social Identity ◽

Social Identification ◽

Negative Group ◽

Identity Threat ◽

Neutral Group ◽

Threat Perceptions ◽

Social Identity Threat ◽

Identity Threats ◽

Group Information ◽

Source Study

The underlying process of reactions to social identity threat was examined from a defense motivation perspective. Two studies measured respondents’ social identification, after which they read threatening group information. Study 1 compared positive and negative group information, attributed to an ingroup or outgroup source. Study 2 compared negative and neutral group information to general negative information. It was expected that negative group information would induce defense motivation, which reveals itself in biased information processing and in turn affects the evaluation of the information. High identifiers should pay more attention to, have higher threat perceptions of, more defensive thoughts of, and more negative evaluations of negative group information than positive or neutral group information. Findings generally supported these predictions.

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Pseudo-Anosov Theory

10.23943/princeton/9780691147949.003.0015 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Braid Groups ◽

Intersection Numbers ◽

Branched Covers ◽

Algebraic Integers ◽

Dehn Twists ◽

Mapping Classes ◽

Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

International Mathematics Research Notices ◽

10.1093/imrn/rnaa290 ◽

2020 ◽

Author(s):

Heather M Russell ◽

Julianna Tymoczko

Keyword(s):

Symmetric Group ◽

Partial Order ◽

Transition Matrix ◽

Group Representation ◽

Jones Polynomial ◽

Basis Matrix ◽

Combinatorial Structures ◽

Quantum Invariants ◽

One Dimensional ◽

Link Diagrams

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 $\mathfrak{sl}_3$.

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