Symplectic quandle Method and $SL(2,\mathbb C)$-representations of 2-bridge Knots
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914277617565696 |
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| author | Jo, Kyeonghee Kim, Hyuk |
| author_facet | Jo, Kyeonghee Kim, Hyuk |
| contents | In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general $SL(2,\mathbb{C})$-representations of 2-bridge ``kmot" groups. We introduce a `generalized symplectic quandle structure' corresponding to ($\mathcal{D}_M$, conjugation) for each $M\in\mathbb C\setminus \{0,1,-1\}$, where $\mathcal{D}_M=\{A\in SL(2,\mathbb{C})\mid tr(A)= M+M^{-1} \}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17433 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symplectic quandle Method and $SL(2,\mathbb C)$-representations of 2-bridge Knots Jo, Kyeonghee Kim, Hyuk Geometric Topology 57K14, 57K31 In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general $SL(2,\mathbb{C})$-representations of 2-bridge ``kmot" groups. We introduce a `generalized symplectic quandle structure' corresponding to ($\mathcal{D}_M$, conjugation) for each $M\in\mathbb C\setminus \{0,1,-1\}$, where $\mathcal{D}_M=\{A\in SL(2,\mathbb{C})\mid tr(A)= M+M^{-1} \}$. 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. |
| title | Symplectic quandle Method and $SL(2,\mathbb C)$-representations of 2-bridge Knots |
| topic | Geometric Topology 57K14, 57K31 |
| url | https://arxiv.org/abs/2601.17433 |