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

TL;DR

This paper classifies non-acyclic ${ m SL}_2$-representations of twist knots, explores their relation to $L$-functions and Dehn surgeries, and analyzes their properties over finite fields, revealing new connections in knot theory and number theory.

Contribution

It provides a complete classification of non-acyclic ${ m SL}_2$-representations of twist knots and links these to $L$-functions and Dehn surgeries, introducing new methods and results.

Findings

Non-acyclic ${ m SL}_2(b C)$-representations lie on the line $x=y$ in character variety.

A representation factors through $(-3)$-Dehn surgery iff it lies on $x=y$.

The $L$-functions of universal deformations are explicitly determined over finite fields.

Abstract

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 $\overline{\rho}$ 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.