I would like to have a transparent material like:
What I have so far is
How could I get rid of the Texture Coordinate and Mapping Nodes? (I just did this because I don't know better)
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You have to keep them. UV Mapping is the default texture coordinate for image textures in Cycles, but the generated textures use generated coordinates by default:
Texture coordinate to sample texture at; defaults to Generated texture coordinates if the socket is left unconnected. (http://wiki.blender.org/index.php/Doc:2.6/Manual/Render/Cycles/Nodes/Textures, Wave Texture)
You could simplify the setup if you pressed CtrlG to make the two nodes a group (a group looks like a single node). You can edit a group by pressing Tab.