Manipulate[
 Show[{r[_a, _b, _t] = 
    ParametricPlot3D[{(a + Cos[b*t]) Cos[t], (a + Cos[b*t]) Sin[t], 
      Sin[b*t]}, {t, 1, 4 Pi + 1}, Axes -> False, Boxed -> False], 
   Graphics3D[{PointSize[0.05], Hue[t/(4 Pi) - 1/(4 Pi)], 
     Point[{(a + Cos[b*t]) Cos[t], (a + Cos[b*t]) Sin[t], Sin[b*t]}], 
     Point[{(a + Cos[b*(4 Pi + 2 - t)]) Cos[(4 Pi + 2 - t)], (a + 
          Cos[b*(4 Pi + 2 - t)]) Sin[(4 Pi + 2 - t)], 
       Sin[b*(4 Pi + 2 - t)]}]}]}, 
  PlotRange -> {{-4, 4}, {-4, 4}, {-4, 4}}], {t, 1, 4 Pi + 1}, {a, -3,
   3}, {b, 1.5, 2.5}]