A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example:
Given a number $N$, a planar unknot is not always possible. For example, a planar unknot is not possible if $N = 3$.
Is there a theorem with which we can readily tell whether a planar integer lattice unknot is possible for a given number of vertices? I already understand that a planar unknot is not possible if the given number of vertices is odd or less than 4. Can a PLU is always possible for any even number of vertices which is greater than 4?
For any even number $N$ larger than 4, there is always at least one such unknot. The minimum planar unknot embedded into an integer lattice is a square involving 4 vertices. The length is 4. For each tug move as defined in Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks by Lomonaco et. al, the length is increased by $2$. So, there is always an unknot for any even number larger than 4.