Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary (the peripheral subgroup) to get a map $\pi_1(\partial N_\epsilon K)\to\operatorname{PSL}(2,\mathbb C)$. Now $\pi_1(\partial N_\epsilon K)$ is just $\mathbb Z\oplus\mathbb Z$, with two generators $m$ (meridian) and $\ell$ (longitude). The image of $\pi_1(\partial N_\epsilon K)$ is parabolic, and so (up to overall conjugation) is of the form: $$ am+b\ell\mapsto\left(\begin{matrix}1&a+b\lambda\cr 0&1\end{matrix}\right) $$ for some complex number $\lambda\in\mathbb C\setminus\mathbb R$.

What is known about this invariant $\lambda(K)$ of hyperbolic knots? Has anyone defined/studied it?