论文标题
non-acyclic $ {\ rm sl} _2 $ - 扭曲结，$ -3 $ -DEHN手术和$ L $ - 功能
Non-acyclic ${\rm SL}_2$-representations of twist knots, $-3$-Dehn surgeries, and $L$-functions
论文作者
论文摘要
我们研究不可约$ {\ rm sl} _2 $ - 扭曲结的代表。我们首先确定所有非酰基$ {\ rm sl} _2（\ mathbb {c}）$ - 表示，这些表示将其置于$ \ mathbb {r}^2 $中的$ x = y $的行上。我们的主要工具是角色品种，雷德扭转和Chebyshev多项式。我们还验证了某种常见的切线特性，该特性会产生通用变形的$ l $ functions，即相关结模块的顺序。其次，我们证明了$ x = y $在且仅当它通过$（-3）$ - dehn手术的因素上的$ x = y $的行为，并且仅当某个元素的图像是第3阶的图像时，并且仅当一个命令3的图像时。 CDVR上通用变形的$L_ρ$。我们还显示$l_ρ$ $ \ dot {=} $ $ $ k_n（x）^2 $在多项式的某个系列$ k_n（x）$中保留。
We study irreducible ${\rm SL}_2$-representations of twist knots. We first determine all non-acyclic ${\rm SL}_2(\mathbb{C})$-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 the $L$-functions of universal deformations, that is, the orders of the associated knot modules. 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 $\overlineρ$ over a finite field with characteristic $p>2$ to concretely determine all non-trivial $L$-functions $L_ρ$ of the universal deformations over a CDVR. We show among other things that $L_ρ$ $\dot{=}$ $k_n(x)^2$ holds for a certain series $k_n(x)$ of polynomials.
