TY - GEN

T1 - Quantum knots and knotted zeros

AU - Kauffman, Louis H.

AU - Lomonaco, Samuel J.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In 2001, Michael Berry4 published the paper "Knotted Zeros in the Quantum States of Hydrogen" in Foundations of Physics. In this paper we show how to place Berry's discovery 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 R3 to the complex plane and such that the inverse image of 0 in the complex plane contains a knotted curve in R3. We show that for knots in R3 this is a generic situation in that every smooth knot K in R3 has a smooth classifying map f: R3-→ C (the complex plane) such that f-1(0) = 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 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.

AB - In 2001, Michael Berry4 published the paper "Knotted Zeros in the Quantum States of Hydrogen" in Foundations of Physics. In this paper we show how to place Berry's discovery 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 R3 to the complex plane and such that the inverse image of 0 in the complex plane contains a knotted curve in R3. We show that for knots in R3 this is a generic situation in that every smooth knot K in R3 has a smooth classifying map f: R3-→ C (the complex plane) such that f-1(0) = 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 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.

KW - Ambient group

KW - Braids

KW - Classifying map

KW - Fibration

KW - Fundamental group

KW - Graphs

KW - Groups

KW - Hamiltonian

KW - Knot complement

KW - Knots

KW - Link of singularity

KW - Links

KW - Quantum computing

KW - Quantum knots

KW - Schrodinger equation

KW - Unitary transformation

KW - quantum knots

KW - link of singularity

KW - classifying map

KW - graphs

KW - unitary transformation

KW - LINK

KW - links

KW - fibration

KW - knot complement

KW - fundamental group

KW - braids

KW - groups

KW - ambient group

KW - quantum computing

KW - knots

UR - http://www.scopus.com/inward/record.url?scp=85068688863&partnerID=8YFLogxK

U2 - 10.1117/12.2518685

DO - 10.1117/12.2518685

M3 - Conference contribution

AN - SCOPUS:85068688863

T3 - Proceedings of SPIE - The International Society for Optical Engineering

BT - Quantum Information Science, Sensing, and Computation XI

A2 - Donkor, Eric

A2 - Hayduk, Michael

A2 - Frey, Michael R.

A2 - Lomonaco, Samuel J.

A2 - Myers, John M.

PB - SPIE

T2 - Quantum Information Science, Sensing, and Computation XI 2019

Y2 - 18 April 2019

ER -