# What is the complex structure on the boundary torus of a hyperbolic knot complement?

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?

• Yes, many people have studied this. I believe it's commonly referred to as the "cusp shape". I imagine Ian Agol will come along and have something to say, but it's a standard thing to study. Google "hyperbolic knot cusp shape" and you'll get plenty of relevant papers. – Ryan Budney Aug 5 '11 at 2:28
• I believe it's known that the meridian of the knot is usually (or perhaps always) the shortest curve in the cusp. – Ryan Budney Aug 5 '11 at 2:30
• FWIW I just asked Morwen Thistlethwaite about this, and he told me there are examples where the meridian isn't shortest – Jim Conant Aug 5 '11 at 3:11
• @Jim: Do they appear in one of the reasonably small censi of knots? If so, it shouldn't take long to find it with a python script. – Ryan Budney Aug 5 '11 at 4:09
• @Ryan - According to SnapPy, the knot K5_15 in the census CensusKnots has merdian just a tad longer than the shortest slope. The shortest slope is a lens space filling. On the other hand, there appears to be no such examples (where the meridian isn't shortest) for knots up to 16 crossings. – Sam Nead Jan 1 '14 at 22:36