Liminal ${\rm SL}_2\mathbb{Z}_p$-representations and odd-th cyclic covers of genus one two-bridge knots

TL;DR

The paper establishes a precise criterion for when genus-one two-bridge knot groups admit liminal ${\rm SL}_2{\mathbb{Z}}_{p}$-characters, linking p-adic deformation theory with the arithmetic of odd cyclic covers via the character variety $f_{k,l}(x,y)$ and Hensel lifting. It connects SL2-deformations to GL1-deformations through coloring of the knot's Alexander polynomial and its resultants, with Legendre symbols governing the p-adic liftability. A key result is that a liminal ${\rm SL}_2{\mathbb{Z}}_{p}$-character exists iff either $p=2$ with a mod-8 condition or an odd $p$ satisfies a quadratic-residue condition on $4k^2l^2-kl$, together with a detailed treatment of Lucas-type sequences and Fox–Weber cyclic covers. The work further explores lifting to liminal representations, presents Lucas/Fibonacci-type identities, and outlines open questions about extending to broader classes of knots and deeper number-theoretic connections. Overall, this study highlights a rich interaction between low-dimensional topology, $p$-adic representation theory, and arithmetic invariants of cyclic covers.

Abstract

Let $p$ be a prime number and let $K$ be a genus one two-bridge knot. In the spirit of arithmetic topology, we observe that if $p$ divides the size of the 1st homology group of some odd-th cyclic branched cover of the knot $K$, then its group $π_1(S^3-K)$ admits a liminal ${\rm SL}_2\mathbb{Z}_p$-character, where $\mathbb{Z}_p$ denotes the ring of $p$-adic integers. In addition, we discuss the existence of liminal ${\rm SL}_2\mathbb{Z}_p$-representations and give a remark on a general two-bridge knot. In the course of argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.

Theorems & Definitions (26)

• Theorem 1.1
• Proposition 2.1
• proof : Proof of \ref{['prop.Jredirr']}
• Remark 2.2
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of \ref{['thm.limchar']}