4 added 86 characters in body
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torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    .3 Sin[.1 + t (2.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description here

Now just insert your six points and draw the lines between each pair by this method.

torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    .3 Sin[.1 + t (2.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description here

torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    .3 Sin[.1 + t (2.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description here

Now just insert your six points and draw the lines between each pair by this method.

3 edited body
source | link
torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (12.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (12.5 - .1)]), 
    .3 Sin[.1 + t (12.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description hereenter image description here

torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (1.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (1.5 - .1)]), 
    .3 Sin[.1 + t (1.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description here

torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    .3 Sin[.1 + t (2.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description here

2 added 5 characters in body
source | link
torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (1.95 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (1.95 - .1)]), 
    .3 Sin[.1 + t (1.95 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line]line, Boxed->False, Axes -> None]

enter image description hereenter image description here

torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (.9 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (.9 - .1)]), 
    .3 Sin[.1 + t (.9 - .1)]}, {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line]

enter image description here

torus = ParametricPlot3D[
  {Cos[θ] (1 + .3 Cos[φ]), 
   Sin[θ] (1 + .3 Cos[φ]), 
   .3 Sin[φ]}, 
  {θ, 0, 2 π}, {φ, 0, 2 π},
  PlotStyle -> Opacity[0.3]];

line = ParametricPlot3D[
   {Cos[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (1.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (1.5 - .1)]), 
    .3 Sin[.1 + t (1.5 - .1)]}, 
    {t, 0, 1}, 
 PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed->False, Axes -> None]

enter image description here

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