  This paper is a memory of the work and influence of Vaughan Jones. It is an
exposition of the remarkable breakthroughs in knot theory and low dimensional
topology that were catalyzed by his work. The paper recalls the inception of
the Jones polynomial and the author's discovery of the bracket polynomial model
for the Jones polynomial. We then describe some of the developments in knot
theory that were inspired by the Jones polynomial and involve variations and
generalizations of this invariant. The paper is written in the form of a
personal odyssey and with the intent to show different mathematical themes that
arise in relation to the Jones polynomial. This invariant can be interpreted in
relation to combinatorial topology, statistical mechanics, Lie algebras, Hopf
algebras, quantum field theory, category theory and more. In each case the
Jones invariant appears as a key example for patterns and connections of these
mathematical and physical contexts. It is remarkable to what extent Vaughan
Jones' discovery of his polynomial has touched so much mathematics, physics,
natural science and so many of our mathematical lives.
