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We explore under what conditions one can obtain a nontrivial knot, given acollection of $n$ vectors. First, we show how to get a crossing from any 3vectors equal in magnitude, by arbitrarily picking 2 vectors and identifyingthe sufficient and necessary criteria for picking a third vector that willguarantee a crossing when the vectors are reordered. We also show that it'salways possible for a set of vectors to be reordered to form the unknot, ifthey sum to $\vec{0}$ when added together.  Our main results are restricted to sets of $n$ vectors that, when reorderedappropriately, 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 nontrivialknot possible ($3_1$) is when $n=7$, and the first $4_1$ knot possible is when$n=8$. We prove that if $n\geq7$, we can always reorder the vectors to get aprojection of a nontrivial knot, and also provide an algorithm to choose how toreorder the vectors to get such a knot.

Conditions for Obtaining Nontrivial Knots from Collections of Vectors