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Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots.

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Τίτλος: Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots.
Συγγραφείς: Jo, Kyeonghee1 (AUTHOR) khjo@mmu.ac.kr, Kim, Hyuk2 (AUTHOR) hyukkim@snu.ac.kr
Πηγή: Journal of Knot Theory & Its Ramifications. Sep2026, Vol. 35 Issue 10, p1-42. 42p.
Θεματικοί όροι: *Knot theory, *Representations of groups (Algebra), *Symbolic computation, *Polynomials
Περίληψη: In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general S L (2 , ℂ) -representations of 2-bridge "kmot" groups. We introduce a 'generalized symplectic quandle structure' corresponding to ( M , conjugation) for each M ∈ ℂ \ { 0 , 1 , − 1 } , where M = { A ∈ S L (2 , ℂ) | t r (A) = M + M − 1 }. By converting the system of conjugation quandle equations to that of generalized symplectic quandle equations, we obtain a simpler expression for the 2-variable Riley polynomial and derive some recursive formulas for Riley polynomials and Alexander polynomials. This approach enables us to effectively compute the A-polynomials, allowing us to obtain numerous previously unknown A-polynomials within minutes using Mathematica. [ABSTRACT FROM AUTHOR]
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  Data: Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots.
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  Data: <searchLink fieldCode="AR" term="%22Jo%2C+Kyeonghee%22">Jo, Kyeonghee</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> khjo@mmu.ac.kr</i><br /><searchLink fieldCode="AR" term="%22Kim%2C+Hyuk%22">Kim, Hyuk</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> hyukkim@snu.ac.kr</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Knot+Theory+%26+Its+Ramifications%22">Journal of Knot Theory & Its Ramifications</searchLink>. Sep2026, Vol. 35 Issue 10, p1-42. 42p.
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  Data: *<searchLink fieldCode="DE" term="%22Knot+theory%22">Knot theory</searchLink><br />*<searchLink fieldCode="DE" term="%22Representations+of+groups+%28Algebra%29%22">Representations of groups (Algebra)</searchLink><br />*<searchLink fieldCode="DE" term="%22Symbolic+computation%22">Symbolic computation</searchLink><br />*<searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general S L (2 , ℂ) -representations of 2-bridge "kmot" groups. We introduce a 'generalized symplectic quandle structure' corresponding to ( M , conjugation) for each M ∈ ℂ \ { 0 , 1 , − 1 } , where M = { A ∈ S L (2 , ℂ) | t r (A) = M + M − 1 }. By converting the system of conjugation quandle equations to that of generalized symplectic quandle equations, we obtain a simpler expression for the 2-variable Riley polynomial and derive some recursive formulas for Riley polynomials and Alexander polynomials. This approach enables us to effectively compute the A-polynomials, allowing us to obtain numerous previously unknown A-polynomials within minutes using Mathematica. [ABSTRACT FROM AUTHOR]
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    Identifiers:
      – Type: doi
        Value: 10.1142/S021821652650015X
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 42
        StartPage: 1
    Subjects:
      – SubjectFull: Knot theory
        Type: general
      – SubjectFull: Representations of groups (Algebra)
        Type: general
      – SubjectFull: Symbolic computation
        Type: general
      – SubjectFull: Polynomials
        Type: general
    Titles:
      – TitleFull: Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots.
        Type: main
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      – PersonEntity:
          Name:
            NameFull: Jo, Kyeonghee
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          Name:
            NameFull: Kim, Hyuk
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          Dates:
            – D: 01
              M: 09
              Text: Sep2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 02182165
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              Value: 35
            – Type: issue
              Value: 10
          Titles:
            – TitleFull: Journal of Knot Theory & Its Ramifications
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