Cubulated moves and discrete knots

Abstract

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.

arXiv comment40 pages, 31 figures