Date of Award
Capstone - Open Access
Mathematics, Minor in Finance
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 ~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 R 2 . We prove that if n = 6, we cannot form a nontrivial knot with our vectors. The first nontrivial knot possible (31) is when n = 7, and the first 41 knot possible is when n = 8. We prove that if n ≥ 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.
Borgatti, Joseph, "Conditions for Obtaining Nontrivial Knots from Collections of Vectors" (2017). Honors Senior Capstone Projects. 12.