Cubulated moves and discrete knots

TL;DR

This paper establishes that cubic knots in three-dimensional space are isotopic if and only if they can be transformed into each other through a finite sequence of cubulated moves, akin to Reidemeister moves, using lattice permutations.

Contribution

It introduces cubulated moves as a discrete analogue to classical knot isotopy and characterizes cubic knots via cyclic permutations of lattice vertices.

Findings

Cubic knots are determined by cyclic permutations of lattice vertices.

Isotopy between cubic knots can be achieved through finite cubulated moves.

Classic knot invariants can be described in terms of lattice permutations.

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.