  The Fermi-Bose mapping method for one-dimensional (1D) Bose and Fermi gases
with zero-range interactions is generalized to an anyon-fermion mapping and
applied to exact solution of several models of ultracold gases with anyonic
exchange symmetry in tight waveguides: anyonic Calogero-Sutherland model,
anyons with point hard core interaction ("anyonic TG gas"), and spin-aligned
anyon gas with infinite zero-range odd-wave attractions ("anyonic FTG gas"). It
is proved that for even $N\ge 4$ there are states of the anyonic FTG gas on a
ring, with anyonic phase slips which are odd integral multiples of $\pi/(N-1)$,
of energy lower than that of the corresponding fermionic ground state. A
generalization to a spinor Fermi gas state with anyonic symmetry under purely
spatial exchange enables energy lowering by the same mechanism.
