Lattice Diagrams of Knots and Diagrams of Lattice Stick Knots
Margaret Allardice, Ethan D. Bloch

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 lattice is not necessarily the projection of a lattice stick knot or link in the lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the lattice is in fact the projection of a lattice stick knot or link.
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Taxonomy
TopicsGeometric and Algebraic Topology · Computational Geometry and Mesh Generation · Image Processing and 3D Reconstruction
