2016/12/20 by Joseph Borgatti, Borgatti, Joseph
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1612.06786
22 pages, 17 figures
arxiv created 2016/12/20 · arxiv updated 2016/12/21
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 ℝ2. 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.