Editors' Note: William Bernard Raymond Lickorish is a Fellow of Pembroke College and Professor of Mathematics at the University of Cambridge, where he was a student. During a short stay between his student years and his return to Cambridge, he taught at the University of Sussex. At Cambridge he was head of the Department of Pure Mathematics and Mathematical Statistics, 1997–2002. A topologist, he has published a book on the topic of this paper, An Introduction to Knot Theory (Springer, 1997). He was awarded the Whitehead Prize by the London Mathematical Society.

Kenneth Cary Millett did his undergraduate work at MIT and took his PhD at the University of Wisconsin-Madison in 1964. He is Professor of Mathematics at the University of California, Santa Barbara, an institution he joined in 1969. The American Mathematical Society gave him their Award for Distinguished Public Service in 1998.

This piece from the Magazine is a result of a conversation over tea one summer in the Mathematics Commons Room at Cambridge when Professor Millett was returning home after a yearlong sabbatical at the Institut des Hautes Études Scientifiques outside Paris. Lickorish and Millett discovered a similarity between the new (at that time) Jones polynomials and the earlier Alexander polynomials used in the study of knots.

For this article the authors were given an Allendoerfer Award in 1989 and this was followed by the Chauvenet Prize in 1991.