TITLE: What is the complex structure on the boundary torus of a hyperbolic knot complement?
QUESTION [8 upvotes]: 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?

REPLY [11 votes]: The conformal structure on the cuspidal torus is usually called the "cusp shape."
See Adams, Hildebrand, Weeks Hyperbolic invariants of knots and links and McReynolds, Arithmetic cusp shapes are dense, for starters.  
See also work on "geometric inflexibility" by Neumann and Reid.
Also check out the work of Marc Lackenby and Jessica Purcell.