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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?

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    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Commented Aug 5, 2011 at 2:28
  • $\begingroup$ I believe it's known that the meridian of the knot is usually (or perhaps always) the shortest curve in the cusp. $\endgroup$
    – Ryan Budney
    Commented Aug 5, 2011 at 2:30
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    $\begingroup$ FWIW I just asked Morwen Thistlethwaite about this, and he told me there are examples where the meridian isn't shortest $\endgroup$
    – Jim Conant
    Commented Aug 5, 2011 at 3:11
  • $\begingroup$ @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. $\endgroup$
    – Ryan Budney
    Commented Aug 5, 2011 at 4:09
  • $\begingroup$ @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. $\endgroup$
    – Sam Nead
    Commented Jan 1, 2014 at 22:36

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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.

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