Couldn't accomplish this in a comment to Rahul's #1, but here is a physical realization of my drawing with colored pencils. Compare:

![alt text][1]
![alt text][2]

The pencils are in contact, but the lead inside could serve as lines in space without contacts.
I don't know how to reconcile this with your quote from Whiteley...


  [1]: http://cs.smith.edu/~orourke/MathOverflow/weaving4.jpg
  [2]: http://cs.smith.edu/~orourke/MathOverflow/pencils4.jpg