  Strongly interacting one-dimensional fermions form an effective spin chain in
the absence of an external lattice potential. We show that the exchange
coefficients of such a chain may be locally tuned by properly tailoring the
transversal confinement. In particular, in the vicinity of a
confinement-induced resonance (CIR) the exchange coefficients may have
simultaneously opposite ferromagnetic and antiferromagnetic characters at
different locations along the trap axis. Moreover, the local exchanges may be
engineered to induce avoided crossings between spin states at the CIR, and
hence a ramp across the resonance may be employed to create different spin
states and to induce spin dynamics in the chain. We show that such unusual spin
chains have already been realized in the experiment of Murmann et al. [Phys.
Rev. Lett. 115, 215301 (2015)].
