  We study the reconnection of vortices in a quantum fluid with a roton
minimum, by numerically solving the Gross-Pitaevskii (GP) equations. A
non-local interaction potential is introduced to mimic the experimental
dispersion relation of superfluid $^4\mathrm{He}$. We begin by choosing a
functional shape of the interaction potential that allows to reproduce in an
approximative way the so-called roton minimum observed in experiments, without
leading to spurious local crystallization events. We then follow and track the
phenomenon of reconnection starting from a set of two perpendicular vortices. A
precise and quantitative study of various quantities characterizing the
evolution of this phenomenon is proposed: this includes the evolution of
statistics of several hydrodynamical quantities of interest, and the
geometrical description of a observed helical wave packet that propagates along
the vortex cores. Those geometrical properties are systematically compared to
the predictions of the Local Induction Approximation (LIA), showing
similarities and differences. The introduction of the roton minimum in the
model does not change the macroscopic properties of the reconnection event but
the microscopic structure of the vortices differs. Structures are generated at
the roton scale and helical waves are evidenced along the vortices. However,
contrary to what is expected in classical viscous or inviscid incompressible
flows, the numerical simulations do not evidence the generation of structures
at smaller or larger scales than the typical atomic size.
