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

Abstract Let K be a genus one two-bridge knot. Let p be a prime number and let ${\mathbb {Z}}_{p}$ denote the ring of p -adic integers. In the spirit of arithmetic topology, we observe that if $p\neq 2$ and p divides (or $p=2$ and $2^3$ 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 $\mathrm { SL}_2{\mathbb {Z}}_p$ -character. In addition, we discuss the existence of liminal $\mathrm {SL}_2{\mathbb {Z}}_{p}$ -representations and give a remark on a general two-bridge knot. In the course of the argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.