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?



What is the complex structure on the boundary torus of a hyperbolic knot complement?
The conformal structure on the cuspidal torus is usually called the "cusp shape."


See Adams, Hildebrand, Weeks [Hyperbolic invariants of knots and links](https://www.jstor.org/stable/2001854 "Trans. Am. Math. Soc. 326, No. 1, 1-56 (1991), doi:10.2307/2001854. zbMATH review at https://zbmath.org/0733.57002") and McReynolds, [Arithmetic cusp shapes are dense](https://doi.org/10.1007/s10711-007-9192-2 "Geom. Dedicata 129, 47-55 (2007), arXiv:math/0606508 [math.GT]. zbMATH review at https://zbmath.org/1143.57010"), for starters.


See also work on "geometric inflexibility" by Neumann and Reid.


Also check out the work of Marc Lackenby and Jessica Purcell.


