  Traditional anyons in two dimensions have generalized exchange statistics
governed by the braid group. By analyzing the topology of configuration space,
we discover that an alternate generalization of the symmetric group governs
particle exchanges when there are hard-core three-body interactions in
one-dimension. We call this new exchange symmetry the traid group and
demonstrate that it has abelian and non-abelian representations that are
neither bosonic nor fermionic, and which also transform differently under
particle exchanges than braid group anyons. We show that generalized exchange
statistics occur because, like hard-core two-body interactions in two
dimensions, hard-core three-body interactions in one dimension create defects
with co-dimension two that make configuration space no longer simply-connected.
Ultracold atoms in effectively one-dimensional optical traps provide a possible
implementation for this alternate manifestation of anyonic physics.
