Conditions for Obtaining Nontrivial Knots from Collections of Vectors

We explore under what conditions one can obtain a nontrivial knot, given a collection of $n$ vectors. First, we show how to get a crossing from any 3 vectors equal in magnitude, by arbitrarily picking 2 vectors and identifying the sufficient and necessary criteria for picking a third vector that will guarantee a crossing when the vectors are reordered. We also show that it’s always possible for a set of vectors to be reordered to form the unknot, if they sum to $\vec{0}$ when added together. Our main results are restricted to sets of $n$ vectors that, when reordered appropriately, project to a regular $n$-gon in ${\mathbb{R}}^2$. We prove that if $n=6$, we cannot form a nontrivial knot with our vectors. The first nontrivial knot possible ($3_1$) is when $n=7$, and the first $4_1$ knot possible is when $n=8$. We prove that if $n\ge 7$, we can always reorder the vectors to get a projection of a nontrivial knot, and also provide an algorithm to choose how to reorder the vectors to get such a knot.