We show how to lift Lagrangian immersions in
to produce
Lagrangian cones in
,
and use this process to produce several families of examples of Lagrangian
cones and special Lagrangian cones. As an application of this theorem, for
we show how
to produce Lagrangian cones that are isotopic to the Harvey–Lawson special Lagrangian
cone and the trivial cone. The projections of the Legendrian links of both of these cones
to
are immersions with four and seven transverse double points. We
expect that these double points represent the chord generators of the
-filtration
level of a suitably defined version of Legendrian contact homology of the
links.