Non-acyclic ${\mathrm{SL}}_2$-representations of twist knots, $-3$-Dehn surgeries, and $L$-functions

Abstract

We study irreducible ${\mathrm{SL}}_2$-representations of twist knots. We first determine all non-acyclic ${\mathrm{SL}}_2(\mathbb{C})$-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 $\overline{\rho }$ over a finite field with characteristic $p>2$ to concretely determine all non-trivial $L$-functions $L_{\rho }$ of the universal deformations over a CDVR. We show among other things that $L_{\rho }$ $\dot{=}$ $k_n(x{)}^2$ holds for a certain series $k_n(x)$ of polynomials.

32 pages, 6 figures; minor modifications in version 4, to appear in Int. Math. Res. Not. IMRN