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Pointed racks and their applications to braid theory.

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Τίτλος: Pointed racks and their applications to braid theory.
Συγγραφείς: Apollos, Angel1 (AUTHOR) aapoll2@lsu.edu, Ceniceros, Jose2 (AUTHOR) jcenicer@hamilton.edu
Πηγή: Journal of Knot Theory & Its Ramifications. Aug2026, p1. 17p.
Θεματικοί όροι: *Braid group (Knot theory), *Representations of groups (Algebra), *Matrices (Mathematics), *Representation theory, *Knot theory
Περίληψη: We introduce a new algebraic structure called a pointed rack and use it to explicitly construct linear representations of the braid group Bn. By tracking the propagation of colorings through braid crossings, we define a rack counting matrix that acts as a set-theoretic permutation representation on the space of rack colorings. We prove that this matrix invariant not only recovers the classical rack coloring invariant of the braid’s closure but also captures strictly finer off-diagonal structural information of the braid. We demonstrate this by distinguishing braids whose closures have the same classical rack invariant. [ABSTRACT FROM AUTHOR]
Βάση Δεδομένων: Academic Search Index
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IllustrationInfo
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  Data: Pointed racks and their applications to braid theory.
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  Data: <searchLink fieldCode="AR" term="%22Apollos%2C+Angel%22">Apollos, Angel</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> aapoll2@lsu.edu</i><br /><searchLink fieldCode="AR" term="%22Ceniceros%2C+Jose%22">Ceniceros, Jose</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> jcenicer@hamilton.edu</i>
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  Data: *<searchLink fieldCode="DE" term="%22Braid+group+%28Knot+theory%29%22">Braid group (Knot theory)</searchLink><br />*<searchLink fieldCode="DE" term="%22Representations+of+groups+%28Algebra%29%22">Representations of groups (Algebra)</searchLink><br />*<searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br />*<searchLink fieldCode="DE" term="%22Representation+theory%22">Representation theory</searchLink><br />*<searchLink fieldCode="DE" term="%22Knot+theory%22">Knot theory</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: We introduce a new algebraic structure called a <italic>pointed rack</italic> and use it to explicitly construct linear representations of the braid group Bn. By tracking the propagation of colorings through braid crossings, we define a rack counting matrix that acts as a set-theoretic permutation representation on the space of rack colorings. We prove that this matrix invariant not only recovers the classical rack coloring invariant of the braid’s closure but also captures strictly finer off-diagonal structural information of the braid. We demonstrate this by distinguishing braids whose closures have the same classical rack invariant. [ABSTRACT FROM AUTHOR]
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1142/s0218216526500446
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 17
        StartPage: 1
    Subjects:
      – SubjectFull: Braid group (Knot theory)
        Type: general
      – SubjectFull: Representations of groups (Algebra)
        Type: general
      – SubjectFull: Matrices (Mathematics)
        Type: general
      – SubjectFull: Representation theory
        Type: general
      – SubjectFull: Knot theory
        Type: general
    Titles:
      – TitleFull: Pointed racks and their applications to braid theory.
        Type: main
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      – PersonEntity:
          Name:
            NameFull: Apollos, Angel
      – PersonEntity:
          Name:
            NameFull: Ceniceros, Jose
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          Dates:
            – D: 20
              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 02182165
          Titles:
            – TitleFull: Journal of Knot Theory & Its Ramifications
              Type: main
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