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

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 ${\mathrm{S}\mathrm{L}}_2{\mathbb{Z}}_p$-character, where ${\mathbb{Z}}_p$ denotes the ring of $p$-adic integers. In addition, we discuss the existence of liminal ${\mathrm{S}\mathrm{L}}_2{\mathbb{Z}}_p$-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.