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A harmonically trapped ultracold 1D spinor Fermi gas with a stronglyattractive 1D even-wave interaction induced by a 3D Feshbach resonance isstudied. It is shown that it has two different super Tonks-Girardeau (sTG)energy eigenstates which are metastable against collapse in spite of the strongattraction, due to their close connection with 1D hard sphere Bose gases whichare highly excited gas-like states. One of these sTG states is a hybrid betweenan sTG gas with strong $(\uparrow\downarrow$ attractions and an ideal Fermi gaswith no $(\uparrow\uparrow)$ or $(\downarrow\downarrow)$ interactions, the sTGcomponent being an exact analog of the recently observed sTG state of a 1Dultracold Bose gas. It should be possible to create it experimentally by asudden switch of the $(\uparrow\downarrow)$ interaction from strongly repulsiveto strongly attractive, as in the recent Innsbruck experiment on the bosonicsTG gas. The other is a trapped analog of a recently predicted sTG state whichis an ultracold gas of strongly bound $(\uparrow\downarrow)$ fermion dimerswhich behave as bosons with a strongly attractive boson-boson interactionleading to sTG behavior. It is proved that the probability of a transition fromthe ground state for strongly repulsive interaction to this dimer state under asudden switch from strongly repulsive to strongly attractive interaction is$\ll 1$, contrary to a previous suggestion.

Two super Tonks-Girardeau states of a trapped 1D spinor Fermi gas