Conditions for Obtaining Nontrivial Knots from Collections of Vectors

TL;DR

This paper investigates conditions under which collections of vectors can be reordered to form nontrivial knots, providing theoretical results and an algorithm for constructing such knots when possible.

Contribution

It establishes necessary conditions for crossing formation, characterizes when nontrivial knots can be formed from vectors, and introduces an algorithm for reordering vectors to produce knots.

Findings

Nontrivial knots cannot be formed with 6 vectors.

The first nontrivial knot (3_1) appears with 7 vectors.

An algorithm is provided to reorder vectors to create nontrivial knots for n ≥ 7.

Abstract

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 $n\geq7$, 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.