  In this paper, we prove than given two cubic knots $K_1$, $K_2$ in
$\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the
other by a finite sequence of cubulated moves. These moves are analogous to the
Reidemeister moves for classical tame knots. We use the fact that a cubic knot
is determined by a cyclic permutation of contiguous vertices of the
$\mathbb{Z}^3$-lattice (with some restrictions), to describe some of the
classic invariants and properties of the knots in terms of such cyclic
permutations, by projecting onto a plane such that it is injective when
restricted to the $\mathbb{Z}^3$-lattice and the image of the
$\mathbb{Z}^3$-lattice is dense.
