  This paper demonstrates the existence of twistless tori and the associated
reconnection bifurcations and meandering curves in the planar circular
restricted three-body problem. Near the Lagrangian equilibrium $\mathcal{L}_4$
a twistless torus is created near the tripling bifurcation of the short period
family. Decreasing the mass ratio leads to twistless bifurcations which are
particularly prominent for rotation numbers 3/10 and 2/7. This scenario is
studied by numerically integrating the regularised Hamiltonian flow, and
finding rotation numbers of invariant curves in a two-dimensional Poincar\'{e}
map.
  To corroborate the numerical results the Birkhoff normal form at
$\mathcal{L}_4$ is calculated to eighth order. Truncating at this order gives
an integrable system, and the rotation numbers obtained from the Birkhoff
normal form agree well with the numerical results. A global overview for the
mass ratio $\mu \in (\mu_4, \mu_3)$ is presented by showing lines of constant
energy and constant rotation number in action space.
