Non-acyclic SL2-representations of Twist Knots, -3-Dehn Surgeries, and L-functions

• Published in: International mathematics research notices Vol. 2022; no. 15; pp. 11690 - 11731
• Main Authors: Tange, Ryoto, Tran, Anh T, Ueki, Jun
• Format: Journal Article
• Language: English
• Published: Oxford University Press 26-07-2022
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnab034

Abstract
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.