Lattice Diagrams of Knots and Diagrams of Lattice Stick Knots

TL;DR

This paper investigates the relationship between lattice diagrams of knots in two dimensions and their realizability as three-dimensional lattice stick knots, providing conditions for when such a projection corresponds to a lattice stick knot.

Contribution

It introduces a necessary and sufficient condition for when a lattice diagram of a knot in ${bZ}^2$ can be realized as a lattice stick knot in ${bZ}^3$, clarifying the connection between 2D diagrams and 3D realizations.

Findings

Identifies that not all ${bZ}^2$ lattice diagrams are projections of ${bZ}^3$ lattice stick knots.

Provides a criterion to determine when a ${bZ}^2$ lattice diagram corresponds to a lattice stick knot.

Clarifies the geometric constraints linking 2D lattice diagrams and 3D lattice knots.

Abstract

We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is in fact the projection of a lattice stick knot or link.