QUANTUM-LIKE PROPERTIES OF KNOTS AND LINKS

Abstract

Knot theory was boosted in its development when in 1860ties Kelvin proposed that knots made out of vortex lines of ether constitute elementary particles of matter. Tait and other associates of Kelvin started to catalogue various types of knots with a hope to create a periodic table of knots which would correspond to periodic tables of chemical elements. However, development of atomic physics, first with classical and then with quantum models of atoms failed to show a connection between knots and atoms. More recently, though, the superstring theory revived the idea that knots can be relevant to understanding elementary particles. Studying ideal geometric configuration of knots we noticed mat their writhe is quantized into units of 4/7 and 10/7 whereby the total writhe of a knot is determined by a simple arithmetic sum of four elementary types of crossings which need to be removed to convert a standard minimal crossing diagram of a given knot into a diagram of a trivial knot without nugatory crossings. There are parallel and antiparallel crossings of positive and negative sign. Interestingly parallel and antiparallel crossings of the same sign show frequently superposition of states. This resembles the quantum properties of elementary particles and their known superposition principle. A model of physical knotted systems is discussed which can change energy states by adopting discrete shapes with associated discrete energies whereby the writhe difference between any two discrete shapes is n*constant.