  The conifold is a basic example of a noncompact Calabi-Yau threefold that
admits a simple flop, and in M-theory, gives rise to a 5d hypermultiplet at low
energies, realized by an M2-brane wrapped on the vanishing sphere. We develop a
novel gauge-theoretic method to construct new classes of examples that
generalize the simple flop to so-called length l=1,...,6. The method allows us
to naturally read off the Gopakumar-Vafa invariants. Although they share
similar properties to the beloved conifold, these threefolds are expected to
admit M2-bound states of higher degree. We demonstrate this through our
computations of the GV invariants. Furthermore we fully characterize the
associated Higgs branches by computing their dimensions and flavor groups. With
our techniques we extract more refined data such as the charges of the hypers
under the flavor group.
