Quantum knots and knotted zeros

Abstract

In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and such that the inverse image of zero in the complex plane contains a knotted curve in three space. We show that for knots in three space this is a generic situation in that every smooth knot K in three space has a smooth classifying map f from three space to the complex plane such that the inverse image of zero is the knot K. This leaves open the question of characterizing just when such f are wave-functions for quantum systems. One can compare this result with the work of Mark Dennis and his collaborators, with the work of Daniel Peralta-Salas and his collaborators, and with the work of Lee Rudolph. Our approach provides great generality to the structure of knotted zeros of a wavefunction and opens up many new avenues for research in the relationships of quantum theory and knot theory. We show how this classifying construction can be related our previous work on two dimensional and three dimensional mosaic and lattice quantum knots.