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

TL;DR

This paper explores the connection between cyclic covers of genus one two-bridge knots and liminal SL2Zp-representations, revealing new links between knot theory, p-adic representations, and number theory.

Contribution

It introduces the concept of liminal SL2Zp-representations for these knots and relates their existence to the divisibility properties of homology groups and Lucas sequences.

Findings

Liminal SL2Zp-representations exist under certain homology divisibility conditions.

Prime divisors of specific Lucas sequences are constrained by Legendre symbols.

Connections between knot covers, p-adic representations, and number theoretic properties are established.

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 $\pi_1(S^3-K)$ 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.