This question is related to this post and its answers:
https://mathoverflow.net/questions/15316/collapsible-group-words

You may think of the three nails as giving a 3-punctured
plane, which has fundamental group a rank 3 free group. 
An element of this group may be thought of as pushing a point
around in the surface. If one takes the trace of this motion
in time, you get a braid with 4 strands, three strands of which
are straight. This represents an element of the pure braid group, an
example of the Birman exact sequence. Closing up the
braid, you get a four component link. Your condition implies 
that removing any two of the last three strands gives the trivial link, so it is 
a kind of second order Brunnian condition.