Academic Journal
Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots.
| Title: | Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots. |
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| Authors: | Jo, Kyeonghee1 (AUTHOR) khjo@mmu.ac.kr, Kim, Hyuk2 (AUTHOR) hyukkim@snu.ac.kr |
| Source: | Journal of Knot Theory & Its Ramifications. Sep2026, Vol. 35 Issue 10, p1-42. 42p. |
| Subject Terms: | *Knot theory, *Representations of groups (Algebra), *Symbolic computation, *Polynomials |
| Abstract: | 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] |
| Database: | Academic Search Index |
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