Was fidgeting with hair-ties, and realized the thing I was trying to make out of them was probably topolgically impossible. Went to the library. Sure enough, on page 83 of Akio Kawauchi's "A Survey of Knot Theory",figure 3.4.4b is a picture of what I was trying to do. They mention that though it it a nontrivial link, it enjoys the property known as Brunnian, or almost trivial, in that every sublink of it is trivial. Which was why I was thought I could build it in the first place, and it would only take one cutting-and-regluing operation to make it.