How to create these famous surfaces in topology with the desired color effects?

Concerning the comment about creating the surfaces, sure: Mathematica is one of the best tools available for that. Here's the Klein bottle, for example.

ParametricPlot3D[{
  (3 + Cos[v/2]*Sin[u] - Sin[v/2]*Sin[2 u])*Cos[v],
  (3 + Cos[v/2]*Sin[u] - Sin[v/2]*Sin[2 u])*Sin[v],
  Sin[v/2]*Sin[u] + Cos[v/2]*Sin[2 u]},
 {u, -Pi, Pi}, {v, 0, 2 Pi},
 Axes -> None,
 ColorFunction -> (Blend[{Purple, Pink, Lighter@Orange}, 
     Mean[{#1, #2, #3}]] &),
 Boxed -> False,
 Mesh -> None,
 PlotStyle -> Directive[Specularity[1, 20], [email protected]]]

Rendering is not Mathematica's strong suit, though. Sure, you could make it do whatever you want if you are willing to write enough code and wait long enough for it to compute, but it would make a lot more sense to model these in Mathematica and then export them in your favorite 3D format for rendering in more appropriate software. My personal preference is Rhino3D.

Edits

Breather surface

From here:

r := 1 - b^2;
w := Sqrt[r];
denom := b*((w*Cosh[b*u])^2 + (b*Sin[w*v])^2)
breather = {-u + (2*r*Cosh[b*u]*Sinh[b*u])/
    denom, (2*w*Cosh[b*u]*(-(w*Cos[v]*Cos[w*v]) - Sin[v]*Sin[w*v]))/
   denom, (2*w*Cosh[b*u]*(-(w*Sin[v]*Cos[w*v]) + Cos[v]*Sin[w*v]))/
   denom}

ParametricPlot3D[
 Evaluate[breather /. b -> 0.4], {u, -13.2, 13.2}, {v, -37.4, 37.4}, 
 PlotRange -> All, PlotPoints -> {60, 150}]

I just finished blog post about the creation of nice graphics from Mathematica Graphics3D using the Blender render framework:

http://wolfig-techblog.blogspot.de/2015/04/blender-as-shader-for-mathematica.html

Maybe you can find some inspiration there for your own graphics. I managed to generate a reasonable Klein bottle with glass shading:

Note: the picture shown here is NOT output created directly with Mathematica, but the result of post-processing with 3D-Studio and afterwards with Blender. The Mathematica code can be found in my blogpost linked above.

If you read through my blog post, you will read as well about the problems I had to get the Klein bottle into Blender. If the readers of this forum have ideas how to overcome these problems I'd be more than happy to get to know about them.

The first picture is the Barth Sextic created in Mathematica with the PlotTheme -> "ThickSurface", the second picture is a "tree" created with the use of AnglePath function