Separating half Giroux torsion does not force the contact invariant to vanish: infinitely many closed counterexamples exist, disproving Ghiggini's conjecture.
Unique holomorphically fillable contact structure on the 3-torus
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Bordered contact invariants and half Giroux torsion
Separating half Giroux torsion does not force the contact invariant to vanish: infinitely many closed counterexamples exist, disproving Ghiggini's conjecture.