We study irreducible $\mathop{\textrm{SL}}\nolimits _2$-representations of twist knots. We first determine all non-acyclic $\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$-representations, which turn out to lie on a line denoted as $x=y$ in ${\mathbb{R}}^2$. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on $L$-functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line $x=y$ if and only if it factors through the $-3$-Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations $\overline{\rho }$ over a finite field with characteristic $p>2$, to concretely determine all non-trivial $L$-functions $L_{{\boldsymbol{\rho }}}$ of the universal deformations over complete discrete valuation rings. We show among other things that $L_{{\boldsymbol{\rho }}}$ $\dot{=}$ $k_n(x)^2$ holds for a certain series $k_n(x)$ of polynomials.

中文翻译：

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扭结、-3-Dehn 手术和 L 函数的非无环 SL2 表示

我们研究了扭结的不可约 $\mathop{\textrm{SL}}\nolimits _2$-表示。我们首先确定所有非循环的 $\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$-表示，它们最终位于 $x=y$ 中的一条线上{\mathbb{R}}^2$。我们的主要工具是字符多样性、Reidemeister 扭转和切比雪夫多项式。我们还验证了某个公共切线属性，它在 $L$ 函数上产生结果，即与通用变形相关的结模块的阶数。其次，我们证明了一个表示在 $x=y$ 线上当且仅当它通过 $-3$-Dehn 手术进行分解，并且当且仅当某个元素的图像是有序的时它是非循环的3. 最后，我们研究了具有特征 $p>2$ 的有限域上的绝对不可约非循环表示 $\overline{\rho }$，具体确定完整离散估值环上普遍变形的所有非平凡 $L$-函数 $L_{{\boldsymbol{\rho }}}$。我们展示了 $L_{{\boldsymbol{\rho }}}$ $\dot{=}$ $k_n(x)^2$ 适用于某个系列 $k_n(x)$ 多项式。