  On a ring, a single Jordan-Wigner transformation between the Kitaev model and
the spin model suffers redundant degrees of freedom. However, we can establish
an exact quaternary Jordan-Wigner mapping involving two Kitaev rings and two
spin rings with periodic or antiperiodic boundary conditions. This mapping
facilitates us to demonstrate exactly how a topological extended-kink (TEK)
phase develops in the interacting Kitaev ring with odd number of lattice sites.
The emergence of this new phase is attributed to the effect of geometrical ring
frustration. Unlike the usual topological phases protected by energy gap in
noninteracting systems, the TEK phase is gapless. And because the spectra of
low energy excitations are quadratic, the specific heat per site approaches a
half of Boltzmann constant near absolute zero temperature. More interestingly,
the ground state is unique, immune to spontaneous symmetry breaking. It
exhibits a long-range correlation function with a nonlocal factor, but no local
order parameter can be defined. As a concomitant effect, a special kind of
localized kink zero mode (KZM) takes place if we introduce a type of bond
defect. We also show that the KZM is robust against moderate disorders.
