The conformal structure on the cuspidal torus is usually called the "cusp shape."

See Adams, Hildebrand, Weeks [Hyperbolic invariants of knots and links][1] and McReynolds, [Arithmetic cusp shapes are dense][2], for starters.  

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

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




  [1]: 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"
  [2]: 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"