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We define an integer-valued virtual count of embedded pseudo-holomorphiccurves of two times a primitive homology class and arbitrary genus insymplectic Calabi--Yau $3$-folds, which can be viewed as an extension ofTaubes' Gromov invariant. The construction depends on a detailed study ofbifurcations of moduli spaces of embedded pseudo-holomorphic curves which ispartially motivated by Wendl's recent solution of Bryan--Pandharipande'ssuper-rigidity conjecture.

Bifurcations of embedded curves and towards an extension of Taubes' Gromov invariant to Calabi-Yau 3-folds