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Motivated by the Double Gyroid nanowire network we develop methods to detectDirac points and classify level crossings, aka. singularities in the spectrumof a family of Hamiltonians. The approach we use is singularity theory. Usingthis language, we obtain a characterization of Dirac points and also show thatthe branching behavior of the level crossings is given by an unfolding of $A_n$type singularities. Which type of singularity occurs can be read off acharacteristic region inside the miniversal unfolding of an $A_k$ singularity.We then apply these methods in the setting of families of graph Hamiltonians,such as those for wire networks. In the particular case of the Double Gyroid weanalytically classify its singularities and show that it has Dirac points. Thisindicates that nanowire systems of this type should have very special physicalproperties.

Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid