This question is related to this post and its answers: 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.

This question is related to this post and its answers: 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.