Mechanical Normal Forms of Knots and Flat Knots

TL;DR

This paper introduces a novel physical knot energy using a resilient metallic tube that naturally finds a unique normal form, outperforming previous energies, and extends this concept to flat knots with applications in knot recognition.

Contribution

It presents a new physical knot energy model that achieves unique normal forms for small crossing knots and flat knots, with theoretical proofs linking knot and flat knot recognition.

Findings

Device performs Reidemeister and Markov moves

Device finds unique normal forms for small crossing knots

Reduces knot recognition to flat knot recognition

Abstract

A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the original knot diagram has a small number of crossings, thus outperforming the famous M\"obius energy due to Jun O'Hara and studied by Michael Freedman et al. Various properties of the device are described (under certain conditions it does the Reidemeister and Markov moves, it beautifully performs the Whitney trick by uniformizing its own local curvature). If the device is constrained between two parallel planes (e.g. glass panes), it yields a real life model of a flat knot (class of knot diagrams equivalent under Reidemeister $\Omega_2$ and $\Omega_3$ moves) also leading to uniquely determined "flat normal forms" (for a small number of crossing points of the given flat knot diagram). The paper concludes with two mathematical theorems, one reducing the knot recognition problem to the flat knot recognition problem, the other (due to S.V.Matveev) giving an easily computable complete system of invariants for the flat unknot knot equivalence problem.