Question about the unknot and grid in $\Bbb R^3$

Solution 1:

This puzzle appeared in Mathematical Mind-Benders, by Peter Winkler, a collection of the interesting mathematical puzzles. Your puzzle (without the grid restriction) was in the chapter "Severe Challenges," and most of the puzzles were hard enough already! There is an excellent discussion of the history of the problem in that book, and also in the introduction of paper: https://arxiv.org/pdf/1507.02355.pdf.

Anyways, here is the simplest solution, which appears on the cover of the Winkler book.

Even more interesting, this blog gives a solution where the embedding is a trefoil knot: